Probability Theory and Mathematical Statistics for Engineers
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Paolo L. Gatti

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Probability Theory and Mathematical Statistics for Engineers

This book is essential reading for practising engineers who need a sound background knowledge of probabilistic and statistical concepts and methods of analysis for their everyday work. It is also a useful guide for graduate engineering students and researchers. The theoretical aspects of modern probability theory is the subject of the ﬁrst part of the book. In the second part, it is shown how these concepts relate to the more practical aspects of statistical analyses. Although separated, the two parts of the book are presented in a uniﬁed style and form a wellstructured unity. Moreover, besides discussing a number of fundamental ideas in detail, the author’s approach to the subject matter is particularly useful for the interested reader who wishes to pursue the study of more advanced topics. This book has an unusual combination of topics based on the author’s education as a Nuclear Physicist and his many years of professional activity as a consultant in different ﬁelds of Engineering. Paolo L. Gatti formerly Head of the Vibration Testing and Data Acquisition Division of Tecniter s.r.l., Cassina de’ Pecchi, Milan, Italy, works now as an independent consultant in the ﬁelds of Engineering Vibrations, Statistical Data Analysis and Data Acquisition Systems.

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Probability Theory and Mathematical Statistics for Engineers

Paolo L. Gatti

First published 2005 by Spon Press 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Spon Press 270, Madison Ave, New York, NY 10016 Spon Press is an imprint of the Taylor & Francis Group © 2005 Paolo L. Gatti This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Gatti, Paolo L., 1959– Probability theory and mathematical statistics for engineers / Paolo L. Gatti. – 1st ed. p. cm. – (Structural engineering: mechanics and design) Includes bibliographical references and index. 1. Engineering – Statistical methods. 2. Probabilities. I. Title. II. Series. TA240.G38 2004 620 .001 5192–dc22

ISBN 0-203-30543-4 Master e-book ISBN

ISBN 0-203-34366-2 (Adobe eReader Format) ISBN 0-415-25172-9 (Print Edition)

2004006776

To the dearest friends of many years, no matter how far they may be. To my parents Paola e Remo and in loving memory of my grandmother Maria Margherita. Paolo L. Gatti

Contents

Preface

x

PART I

Probability theory

1

1

3

The concept of probability 1.1 1.2 1.3 1.4 1.5

2

Different approaches to the idea of probability 3 The classical deﬁnition 4 The relative frequency approach to probability 14 The subjective viewpoint 18 Summary 19

Probability: the axiomatic approach

22

2.1 2.2 2.3 2.4

Introduction 22 Probability spaces 22 Random variables and distribution functions 33 Characteristic and moment-generating functions 55 2.5 Miscellaneous complements 63 2.6 Summary and comments 72 3

The multivariate case: random vectors 3.1 3.2 3.3

Introduction 76 Random vectors and their distribution functions 76 Moments and characteristic functions of random vectors 85 3.4 More on conditioned random variables 103

76

viii

Contents 3.5 3.6

4

Functions of random vectors 115 Summary and comments 126

Convergences, limit theorems and the law of large numbers 4.1 4.2 4.3 4.4 4.5 4.6 4.7

129

Introduction 129 Weak convergence 130 Other types of convergence 137 The weak law of large numbers (WLLN) 142 The strong law of large numbers (SLLN) 146 The central limit theorem 149 Summary and comments 157

PART II

Mathematical statistics

161

5

163

Statistics: preliminary ideas and basic notions 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

6

Introduction 163 The statistical model and some notes on sampling 164 Sample characteristics 168 Point estimation 180 Maximum likelihood estimates and some remarks on other estimation methods 196 Interval estimation 204 A few notes on other types of statistical intervals 217 Summary and comments 218

The test of statistical hypotheses

223

6.1 6.2 6.3 6.4

Introduction 223 General principles of hypotheses testing 223 Parametric hypotheses 226 Testing the type of distribution (goodness-of-ﬁt tests) 251 6.5 Miscellaneous complements 262 6.6 Summary and comments 271 7

Regression, correlation and the method of least squares 7.1 7.2

Introduction 274 The general linear regression problem 275

274

Contents 7.3 7.4 7.5

Normal regression 285 Final remarks on regression 298 Summary and comments 303

Appendix A: elements of set theory A.1 A.2 A.3

B.3 B.4 B.5

306

Basic deﬁnitions and properties 306 Functions and sets, equivalent sets and cardinality 312 Systems of sets: algebras and σ -algebras 315

Appendix B: the Lebesgue integral – an overview B.1 B.2

ix

318

Introductory remarks 318 Measure spaces and the Lebesgue measure on the real line 319 Measurable functions and their properties 321 The abstract Lebesgue integral 324 Further results in integration and measure theory and their relation to probability 330

Appendix C

337

C.1 The Gamma Function (x) 337 C.2 Gamma distribution 338 C.3 The χ 2 distribution 339 C.4 Student’s distribution 340 C.5 Fisher’s distribution 342 C.6 Some other probability distributions 344 C.7 A few ﬁnal results 350 Index

353

Preface

Tanta animorum imbecillitas est, ubi ratio decessit L.A. Seneca (De Constantia Sapientis, XVII)

A common attitude among engineers and physicists is, loosely speaking, to consider statistics as a tool in their toolbox. They know it is ‘there’, they are well aware of the fact that it can be of great help in a number of cases and, having a general idea of how it works, they ‘dust it off’ and use it whenever the problem under study requires it. A minor disadvantage of this pragmatic, and in many ways justiﬁable (after all, statistics is not their main ﬁeld of study) point of view is that one often fails to fully appreciate its potential, its richness and the complexity of some of its developments. A more serious disadvantage is the risk of improper use, although it is fair to say that this is rarely the case in science when compared with other ﬁelds of activity such as, for instance, politics, advertising and journalism (even assuming the good faith of the individuals involved). These general considerations aside, in the author’s mind the typical reader of this book (whatever the term ‘typical reader’ may mean) is an engineer or physicist who has a particular curiosity – personal and/or professional – for probability and statistics. Although this typical reader is surely interested in statistical techniques and methods of practical use, his/her focus is on ‘understanding’ rather than ‘information’ on this or that speciﬁc method, and in this light he/she is willing to tackle some mathematical difﬁculties in order to, hopefully, achieve this understanding. Since I found myself in this same situation a few years ago and it is now my opinion that the reward is well-worth the effort (this, however, in no way implies that I have reached a full understanding of the subject-matter – unfortunately, I feel that it is not so – it simply means that after a few years of study the general picture is much clearer and many details are much sharper now than then), I decided to write a book which would have fulﬁlled my needs at that time. It goes without saying that there are many good books on the subject – a good number of them are on my shelf and I have often referred to them either for work or in writing this book – and this is why I included a rather detailed list of references at the end of each chapter.

Preface xi The book is divided into two main parts: Part 1 (Chapters 1–4) on probability theory and Part 2 (Chapters 5–7) on mathematical statistics. In addition, three appendices (A, B and C) complement the book with some extra material relevant to the ideas and concepts presented in the main text. With regard to Part 1 on probability, some mathematical difﬁculties arise from the circumstance that the reader may not be familiar with measure theory and Lebesgue integration, but I believe that it would have been unfair to the ‘typical reader’ to pretend to ignore that modern probability theory relies heavily on this branch of mathematics. In Part 2, on the other hand, I tried as much as possible to show the way in which, in essence, this part is a logical – even if more application-oriented – continuation of the ﬁrst; a fact that, although obvious in general, is sometimes not clear in its details. In all, my main goal has been to give a uniﬁed treatment in the hope of providing the reader with a clear wide-angle picture where, in addition, some important details are in good focus. On this basis, in fact, he/she will be able to pursue the study of more advanced topics and understand the main ideas behind the speciﬁc statistical techniques and methods – some of which are rather sophisticated indeed – that he/she will encounter in this and/or other texts. Also, it is evident that in writing a book like this some compromise must be made on the level of mathematical exposition and selection of topics. In regard to the former I have striven for clarity rather than mathematical rigor; in fact, there exist many excellent books written by mathematicians (some of them are included in the references) where rigor is paramount and all the necessary proofs are given in detail. For the latter, it is only fair to say that many important topics, including probably at least one of everyone’s favourites, have been omitted. Out of necessity, in fact, some choices had to be made (a few of them have been made painfully along the way, leaving some doubts that still surface now and then) and I tried to do so with the intention of writing a not-too-long book without sacriﬁcing the spirit of the original idea that had me started in the ﬁrst place. Only the readers will be able to tell if I have been successful and faithful to this idea. Finally, it is possible that, despite the attention paid to reviewing all the material, this book will contain errors, omissions, oversights or misprints. I will be grateful to the readers who spot any of the above or who have any comments for improving the book. Any suggestion will be received and considered. Paolo L. Gatti Milano September 2004

Part I

Probability theory

1

The concept of probability

1.1

Different approaches to the idea of probability

Probabilistic concepts, directly or indirectly, pervade many aspects of human activities, from everyday situations to more advanced and speciﬁc applications in natural sciences, engineering, economy and politics. It is the scope of this introductory chapter to discuss the fundamental idea of probability which, as we will see, is not so obvious and straightforward as it may seem. In fact – in order to deal with practical problems in a ﬁrst stage and to arrive at a sound mathematical theory later – this concept has evolved through the centuries, changing the theory of probability from an almost esoteric discipline to a well-established branch of mathematics. From a strict historical point of view, despite the fact that some general notions have been common knowledge long before the seventeenth century (e.g. Cardano’s treatise ‘Libel de Ludo Aleæ’ (Book of Dice Games) was published in 1663 but written more than a century earlier), the ofﬁcial birth of the theory dates back to the middle of the seventeenth century and its early developments owe much to great scientists such as Pascal (1623–1662), Fermat (1601–1665), Huygens (1629–1695), J. Bernoulli (1654–1705), de Moivre (1667–1754), Laplace (1749–1827) and Gauss (1777–1855). Broadly speaking, probability is a loosely deﬁned term employed in everyday conversation to indicate the measure of one’s belief in the occurrence of a future event when this event may or may not occur. Moreover, we use this word by indirectly making some common assumptions: probabilities near 1 (100%) indicate that the event is extremely likely to occur, probabilities near zero indicate that the event is almost not likely to occur and probabilities near 0.5 (50%) indicate a ‘fair chance’, that is, that the event is just as likely to occur as not. If we try to be more speciﬁc, we can consider the way in which we assign probabilities to events and note that three main approaches have developed through the centuries. Following the common terminology, we call them (1) the classical approach, (2) the relative frequency approach, (3) the subjective approach.

4

Probability theory

This order agrees with the historical sequence of facts. In fact, the classical deﬁnition of probability was the ﬁrst to be given, followed by the relative frequency deﬁnition and – not long before Kolmogorov’s axiomatic approach was introduced in 1931 – by the subjective deﬁnition. Let us examine them more closely.

1.2

The classical deﬁnition

The ﬁrst two viewpoints mentioned in Section 1.1, namely the classical and the relative frequency approaches, date back to a few centuries ago and originate from practical problems such as games of chance and life insurance policies, respectively. Let us consider the classical approach ﬁrst. In a typical gambling scheme, the game is set up so that there exists a number of possible outcomes which are mutually exclusive and equally likely and the gambler bets against the House on the realization of one of these outcomes. The tossing of a balanced coin is the simplest example: there are two equally likely possible outcomes, head or tail, which are mutually exclusive (that is both faces cannot turn up simultaneously) and the bet is, say, the appearance of a head. More speciﬁcally, the classical (or the gambler’s) deﬁnition of probability can be used whenever it can be reasonably assumed that the possible outcomes of the ‘experiment’ are mutually exclusive and equally likely so that one calculates the probability of a particular outcome A as P(A) =

n(A) n(S)

(1.1)

where n(A) is the number of ways in which outcome A can occur and n(S) is the total number of ways in which the experiment can proceed. Note that with this deﬁnition we do not need to actually perform the experiment because eq. (1.1) deﬁnes an ‘a priori’ probability. In tossing a fair coin, for instance, this means that without even trying we can say that n(S) = 2 (head or tail) and the probability of a head is P(A) ≡ P(head) = 1/2. Also, in rolling a fair die – where six outcomes are possible, that is, n(S) = 6 – the appearance of any one particular number can be calculated by means of eq. (1.1) and gives 1/6 while, on the other hand, the appearance of, say, an even number is 1/2. 1.2.1

Properties of probability on the basis of the classical deﬁnition

In the light of the simple examples given in Section 1.2, the classical deﬁnition can be taken as a starting point to give some initial deﬁnitions and determine a number of properties which we expect a ‘probability function’ to have. This will be of great help in organizing some intuitive notions in a more systematic

The concept of probability

5

manner – although, for the moment, in a rather informal way and on the basis of heuristic considerations (the term ‘probability function’ itself is here used informally just to point out that a probability is something that assigns a real number to each possible outcome of an experiment). In order to do so we must turn to the mathematical theory of sets (the reader may refer to Appendix A for some basic aspects of this theory). First of all, we give some deﬁnitions: (a) we call event a possible outcome of a given experiment; (b) among events, we distinguish between simple events, which can happen only in one way, are mutually exclusive and equally likely; (c) compound events, which can happen in more than one way. Then, (d) we call sample space (or event space) the set of all possible simple events. Note that this deﬁnition justiﬁes the fact that simple events are also often called sample points. In the die-rolling experiment, for example, the sample space is the set {1, 2, 3, 4, 5, 6}, a simple event is the observation of a six and a compound event is the observation of an even number (2, 4, or 6). Adopting the notations of set theory, we can view the sample space as a set W whose elements Ej are the sample points. Then, any compound event A is a subset of W and can be viewed as a collection of two or more sample points, that is, as the union of two or more simple events. In the die-rolling experiment above, for example, we can write A = E2 ∪ E4 ∪ E6

(1.2)

where we called A the event ‘observation of an even number’, E2 the sample point ‘observation of a 2’ and so on. In this case, it is evident that P(E2 ) = P(E4 ) = P(E6 ) = 1/6 and, since E2 , E4 and E6 are mutually exclusive we expect an ‘additivity property’ of the form P(A) = P(E2 ∪ E4 ∪ E6 ) = P(E2 ) + P(E4 ) + P(E6 ) = 1/2

(1.3a)

An immediate consequence of eq. (1.3a) is that ⎛ P(W) = P ⎝

6

j=1

⎞ Ej ⎠ =

6

P(Ej ) = 1

(1.3b)

j=1

because it is clear that one of the six faces must necessarily show up. Moreover, if we denote by AC the complement of set A (clearly W = A ∪ AC : for example, in the die experiment if A is the appearance of an even

6

Probability theory

number then the event AC represents the non-occurrence of A, that is, the appearance of an odd number; therefore AC = E1 ∪ E3 ∪ E5 ), we have P(AC ) = 1 − P(A)

(1.4)

If, on the other hand, we consider two events, say B and C, which are not mutually exclusive, a little thought leads to P(B ∪ C) = P(B) + P(C) − P(B ∩ C)

(1.5a)

where P(B ∩ C) is called the compound probability of events B and C, that is, the probability that B and C occur simultaneously. An example will help clarify this idea: returning to our die-rolling experiment, let, for example, B = E2 ∪ E3 and C = E1 ∪ E3 ∪ E6 , then B ∩ C = E3 and, as expected, P(B ∪ C) = (2/6) + (3/6) − (1/6) = (4/6). For three non-mutually exclusive events, say B, C and D, eq. (1.5a) becomes P(B ∪ C ∪ D) = P(B) + P(C) + P(D) − P(B ∩ C) − P(B ∩ D) − P(C ∩ D) + P(B ∩ C ∩ D)

(1.5b)

as the reader is invited to verify. In general, the extension of eq. (1.5a) to n events A1 , A2 , . . . , An leads to the rather cumbersome expression ⎛ ⎞ n n P⎝ Ak ⎠ = P(Ak ) − P(Ak1 ∩ Ak2 ) + · · · + (−1)m+1 k=1

k=1

×

k1 0) are two real parameters (whose meaning is probably well known to the reader but will be shown later). The fact that the pdf (2.29a) satisﬁes the normalization condition (eq. (2.25b)) can be veriﬁed by writing 1 √ σ 2π

+∞ +∞ 1 (t − x)2 2 dt = √ exp − e−(y /2) dy 2σ 2 2π

−∞

−∞

where the second integral is obtained by the change of variable ¯ . +∞ √ y = (x− x)/σ Since from integrals tables we get −∞ exp(−ax2 ) dx = π/a, eq. (2.25b) follows. The PDF of eq. (2.29a) cannot be written in explicit analytical form but it is given by

1 FX (x) = √ σ 2π

x −∞

(t − x) ¯ 2 exp − 2σ 2

1 dt = √ 2π

(x− ¯ ) x/σ

e−(y

2

/2)

dy

−∞

(2.29b) where, again, the second integral is obtained by the same change of variable as above and – due to its importance in statistics – can be easily found in numerical table form. However, if tables are not available, the approximation 1 F(z) ≡ 2π

z −∞

exp(−y2 /2) dy ∼ =

1 1 + exp{−az(1 + bz2 )}

(2.29c)

with a = 1.5976 and b = 0.044715 is sufﬁciently accurate for most applications (the maximum absolute error is ≤ 2 × 10−4 ). Other examples of absolutely continuous probability laws will be given later. As remarked above, in almost all practical cases the continuous singular part of the decomposition is generally absent. As a consequence, it is customary to speak of mixed random variables when neither the absolutely continuous part nor the discrete part of the PDF function are identically zero.

Probability: the axiomatic approach

45

In this case there exist a number of points xn for which eq. (2.17) holds; however eq. (2.18) is no longer true and we have

pn =

n

P{X −1 (xn )} < 1

(2.30)

n

This means that there exist at least a pair of neighboring points xn and xn+1 such that F(xn ) < F(xn+1 −). In other words, F can be written as the sum (called a ‘convex’ linear combination) F(x) = αFac (x) + (1 − α)Fd (x)

(2.31)

where 0 ≤ α ≤ 1, Fac is an absolutely continuous, monotonically increasing function and Fd is a jump function of the type (2.20). Obviously, α = 1 corresponds to the absolutely continuous case and α = 0 to the discrete case. In both cases – and, clearly, also in the mixed case – we will see in the next section how the Lebesgue–Stieltjes integral is the appropriate tool used to calculate important quantities such as the mean value, the variance and, in general, many other parameters which describe in numerical form the behavior of a random variable. Example 2.5

Suppose a r.v. has the following PDF

⎧ ⎪ xb

called the uniform PDF on [a, b]. This function is absolutely continuous and its pdf is f (x) = 1/(b − a) for x ∈ (a, b) and zero elsewhere. Also, it is immediate to determine that a r.v. X whose PDF is given by (2.57a) has moments given by b E(X k ) =

xk f (x) dx = a

bk+1 − ak+1 (b − a)(k + 1)

(2.57b)

Probability: the axiomatic approach

65

from which it follows E(X) = (a + b)/2 and, with a little algebra, Var(X) = (b2 − a2 )/12. The characteristic function is also obtained with little effort and we get ϕ(u) =

eiub − eiua iu(b − a)

(2.57c)

which gives an indeterminate form 0/0 for u = 0; however ϕ(u) → 1 as u → 0. If a = −b the function F(x) is even, the CF is real and can be written as ϕ(u) = sin(bu)/bu. 2.5.2

More on conditional probability

The second aspect we consider here deals with conditional probabilities. We introduced the concept informally in Section 1.2.1 and made some further comments in Section 2.2. There we pointed out that – given a probability space (W, S, P) and a conditioning event G ∈ S with P(G) > 0 – the set function PG : W → [0, 1] deﬁned (for A ∈ S) by the relation PG (A) = P(A ∩ G)/P(G) is a probability function in its own right which, often, is also denoted by P(A|G). Also, if X : W → R is a r.v. on (W, S, P) we observe that X is a r.v. (i.e. measurable) on the space (W, S, PG ) as well, because – we recall from Deﬁnition 2.3 – the measurability of functions is independent on the measure P. We have now two probability measures on (W, S), that is, P and PG , and the ﬁrst thing to note is that PG is absolutely continuous with respect to P because PG (A) = 0 whenever P(A) = 0. Then, the Radon–Nikodym theorem states that there is an essentially unique function H : W → R such that PG (A) =

H dP A

This function is called the Radon–Nikodym derivative of PG with respect to P and it often symbolically denoted by dPG /dP. We state now that H=

IG P(G)

(2.58)

where IG is the indicator function of the set G. In fact, by the deﬁning properties of abstract Lebesgue integral we have P(A ∩ G) =

dP =

A∩G

IA∩G dP =

W

IA IG dP =

W

IG dP A

66 Probability theory where the third equality holds because it is immediate to prove that IA∩G = IA IG . Substituting this result into the deﬁnition of PG we get for every A ∈ S PG (A) =

1 P(G)

IG dP

(2.59)

A

which, in the light of uniqueness of H, proves eq. (2.58). Given a r.v. X on W, a more interesting – and useful in practice – case is when the conditioning set G is the counterimage through X of a Borel set C ⊂ R, that is, when G = X −1 (C). Then, calling PC the probability measure deﬁned by

PC (A) =

P[A ∩ X −1 (C)] P(X −1 (C))

(2.60)

(Incidentally, this notation may seem strange because PC is a measure in (W, S) but C is a Borel set in the domain of X; rigorously one should write PX−1 (C) but then the notation would become too heavy) we can consider its image measure PX|C in R and note that it is absolutely continuous with respect to PX (the image measure of P). By a similar argument as above, the Radon–Nikodym theorem applies. So – recalling the relation between the abstract Lebesgue integrals in dP and dPX and noting that X −1 (B) ∩ X −1 (C) = X −1 (B ∩ C) – from the chain of equalities

PX|C (B) = PC [X

=

−1

1 PX (C)

P[X −1 (B ∩ C] 1 (B)] = = P(X −1 (C)) P(X −1 (C)) dPX =

B∩C

1 PX (C)

dP X −1 (B∩C)

IC dPX

(2.61)

B

it follows that the Radon–Nikodym derivative dPX|C /dPX is dPX|C IC = dPX PX (C)

(2.62)

If now we turn our attention to the conditional PDF deﬁned as FX|C (x) = PC [X −1 ( Jx )]

(2.63)

Probability: the axiomatic approach

67

where Jx = (−∞, x], we can use the basic result of eq. (2.61) to get FX|C (x) =

1 PX (C)

IC dPX = Jx

1 PX (C)

IC dFX

(2.64)

Jx

where the second integral is a Lebesgue–Stieltjes integral. If, in addition, the function FX is absolutely continuous on R then the pdf fX exists and we can take the derivative of both sides to obtain the conditional pdf in terms of the unconditional one fX|C (x) =

IC fX (x) IC fX (x) = PX (C) fX (x) dx

(2.65a)

C

On the other hand, if FX is discrete we have pX|C (xk ) =

IC pX (xk ) IC pX (xk ) = PX (C) xi ∈C p(xi )

(2.65b)

Equations (2.65a) and (2.65b) show that fX|C (or pX|C ) is zero outside the set C while in C, besides the multiplicative constant 1/PX (C), coincides with the unconditioned pdf (or pmf). The factor 1/PX (C) is necessary in order to satisfy the normalization condition. Using the conditional characteristics we can deﬁne and calculate the conditional moments just as we did in the unconditioned case. So, we can deﬁne E(X k |C) =

X k dPC =

xk dPX|C =

R

W

xk dFX|C

(2.66a)

R

where the second equality holds because of the relation between PC and its image measure PX|C and the third equality holds because of the deﬁnition of Lebesgue–Stieltjes integral. Then, by virtue of eq. (2.62) we also have E(X k |C) =

1 PX (C)

xk dFX

(2.66b)

C

where the integral on the r.h.s. is a sum or a Lebesgue integral on R (which in most practical cases coincides with an ordinary Riemann integral) depending on whether FX is discrete or absolutely continuous. Similarly, we can deﬁne the conditional-CF as ϕX|C (u) = E(eiuX |C) and, more generally, the

68 Probability theory expectation of a (Borel) function g(X). In the absolutely continuous case, for example, we get E[g(X)|C] =

Cg(x)fX (x) dx

(2.67)

C fX (x) dx

Clearly, when all events B ⊂ R are independent of the conditioning event C then the conditioned characteristics coincide with the unconditioned ones; the simplest case of this situation is when C = R or, in the space (W, S), when G = W. We close this section with a ﬁnal result regarding conditional expectations which is somehow a counterpart of the total probability formula of eq. (1.12). This result is one form of the so-called total expectation theorem and is given in the following proposition: Proposition 2.27 (Total expectation theorem) Let the sets Gi ∈ S be such that W = ∪i Gi , Gi ∩Gj = ∅ for i = j and P(Gi ) > 0 for all i = 1, 2, . . . . Then E(X) =

P(Gi )E(X|Gi )

(2.68)

i

In fact, as a consequence of the Radon–Nikodym theorem, we can write E(X|Gi ) = [P(Gi )]−1 Gi X dP but then, owing to the properties of the abstract Lebesgue integral we get

X dP =

E(X) = W

X dP =

! i

Gi

X dP

i G i

so that Proposition 2.26 follows from these two results. For the moment, the considerations above sufﬁce and we leave further developments on conditional probability to future sections. In particular, for continuous random variables we will show how one can condition on an event with zero probability, that is an event of the form X = x0 where x0 is a speciﬁed value and PX {x0 } = P(X −1 {x0 }) = 0. 2.5.3

Functions of random variables

The third topic we consider here deals with random variables with a known functional dependence on another random variable. So, let X be a r.v. with known probability distribution FX and let g : R → R be a well-behaved Borel function. Since we already know that the function Y(w) ≡ g(X(w)) : W → R is itself a random variable, we may ask for its probability distribution.

Probability: the axiomatic approach

69

This is not always simple and we will consider only some frequently encountered cases. Suppose ﬁrst that X is absolutely continuous with pdf fX and g is monotonically increasing. Then, given a value y, we have that Y ≤ y whenever X ≤ x (where y = g(x)) and Y is also absolutely continuous. Moreover, in this case the inverse function g −1 exists is single-valued and g −1 (y) = x; therefore FY (y) = P(Y ≤ y) = P[X ≤ g −1 (y)] = FX (g −1 (y))

(2.69a)

then, taking the derivative with respect to y we obtain the pdf of Y as fY (y) = fX (g −1 (y))

dg −1 (y) dy

(2.69b)

If, on the other hand, g is monotonically decreasing, then we have FY (y) = P(Y ≤ y) = P[X > g −1 (y)] = 1 − P[X ≤ g −1 (y)] = 1 − FX (g −1 (y))

(2.70a)

and differentiating fY (y) = −fX (g −1 (y))

dg −1 (y) dy

(2.70b)

By noting that the derivative dg −1 /dy is positive when g is monotonically increasing and negative when g is monotonically decreasing, eqs (2.69b) and (2.70b) can be combined into the single equation fY (y) = fX (g

−1

dg −1 (y) (y)) dy

(2.71)

Example√2.10(a) Let X be a r.v. with pdf fX = e−x (x ≥ 0) and let Y = 2 g(X) = X. Then x = g −1 (y) = y2 , fX (g −1 (y)) = e−y and dg −1 /dy = 2y, 2 so that, by eq. (2.69b), we get fY (y) = 2ye−y (y ≥ 0). Clearly, if we note 2 that FX (x) = 1 − e−x we can use eq. (2.69a) to get the PDF FY (y) = 1 − e−y , as expected, can also be obtained by computing the integral FY (y) = which, y f (t)dt. As an easy exercise, the reader is invited to sketch a graph of fX Y 0 and fY and note that they are markedly different. Example 2.10(b) Let us now consider the linear case Y = g(X) = aX + b. This is an increasing function for a > 0 and decreasing for a < 0. If fX (x)

70 Probability theory is the pdf of X, eq. (2.71) yields fY (y) = |a|−1 fX ((y − b)/a). Also, FY (y) = FX ((y − b)/a) if a > 0 and FY (y) = 1 − FX ((y − b)/a) if a < 0. So, for example, if the original r.v. X is normally distributed – that is, its pdf is given by eq. (2.29a) – and a > 0 we get

(y − ax¯ − b)2 exp − fY (y) = 2a2 σ 2 aσ 2π 1 √

meaning that Y is a Gaussian r.v. itself with mean y¯ = ax¯ + b and variance σY2 = a2 σ 2 . Example 2.10(c) Starting again from the normal pdf of eq (2.29a) we can obtain the pdf of the standardized normal r.v. Y = (X − x)/σ ¯ . Noting that g −1 (y) = σ y + x and dg −1 /dy = σ we get 1 exp(−y2 /2) fY (y) = √ 2π which, as mentioned at the end of Example 2.8, is a normal r.v. with x¯ = 0 and σ = 1. If g is not monotone, it can often be divided into monotone parts; the considerations above then apply to each part and in the end the sum of the various parts is taken. A simple example of this latter case is Y = g(X) = X 2 which is decreasing for x < 0 and increasing for x > 0. Since g(x) is always positive for all x (or, stated differently, g −1 (y) = ∅ for y < 0) the r.v. Y cannot take on negative values and therefore fY (y) = 0 for y < 0. For y > 0 it is left to the reader to determine that the sum of the two parts leads to √ √ √ fY (y) = (2 y)−1 (fX (− y) + fX ( y)). Example 2.11 In applications it is often of interest to have a probabilistic description of the maximum or minimum of a number n of r.v.s. As we will see in later chapters, an important case is when the r.v.s X1 , X2 , . . . , Xn are independent and have the same PDF F(x). Now, ﬁrst of all it can be shown that the function Y = max{X1 , . . . , Xn } is itself a r.v. (as is the minimum). Then, since FY (y) = P(max{X1 , . . . , Xn } ≤ y) = P(X1 ≤ y, . . . , Xn ≤ y) the assumption of independence leads to FY (y) = P(X1 ≤ y, . . . , Xn ≤ y) =

n

i=1

P(X1 ≤ y) = (F(y))n

(2.72a)

Probability: the axiomatic approach

71

and therefore, if F is absolutely continuous fY (y) = n(F(y))n−1 f (y)

(2.72b)

where f is the derivative of F. We note here that the expression P(X1 ≤ y, . . . , Xn ≤ y) is written in the usual ‘shorthand’ notation of probability theory; in rigorous mathematical symbolism, however, this probability is written P ∩ni=1 Xi−1 (−∞, y] . The rigorous notation is useful when we consider the minimum of the r.v.s X1 , X2 , . . . , Xn . In fact, if Jy = (−∞, y] we have −1 FY (y) = P(min{X1 , . . . , Xn }) = P Xi ( Jy ) "

(X −1 ( Jy ))C =P

#C

i

=1−P

i

"

# (X

−1

( Jy ))

C

i

where in the third equality we used de Morgan’s law. Then, by virtue of $ % & $ %C independence P ∩i (Xi−1 ( Jy ))C = i P Xi−1 ( Jy ) = (1 − F(y))n , so that putting the pieces together we ﬁnally get FY (y) = 1 − (1 − F(y))n

(2.73a)

and if F is absolutely continuous (F = f ) fY (y) = n(1 − F(y))n−1 f (y)

(2.73b)

So, for instance, if F is the uniform PDF (eq. (2.57a)) on the interval [a, b] = [0, 1] then FY (y) = n(1 − y)n−1 where 0 ≤ y ≤ 1. If X is a discrete r.v. whose range is the set AX = {x1 , x2 , . . .} = {xi } then Y = g(X) is also a discrete r.v. because its range is the set AY = {y1 , y2 , . . .} = {yk } (note that the elements of AX and AY are labelled by different indexes because, in general, a given value yk may be the image – through g – of more than one xi ). In this case, in general, it is not convenient to go through the PDF but it is better to determine the mass distribution pY by ﬁrst identifying the values yk and then using the relation pY (yk ) = P(Y = yk ) = P(X = g −1 (yk )) =

pX (g −1 (yk ))

(2.74)

xi

where the sum is taken on all values xi (when there is more than one) which are mapped in yk . So, for instance, let X be such that AX = {x1 = −1,

72 Probability theory x2 = 0, x3 = 1}, pX (−1) = pX (0) = 0.25 and pX (1) = 0.5 and let Y = X 2 . Then AY = {y1 , y2 } = {0, 1} and g −1 (y2 ) = −1 ∪ 1 = x1 ∪ x3 so that in calculating pY (y2 ) we must sum the probabilities pX (x1 ) and pX (x3 ). Therefore pY (y2 ) = 0.75. By contrast, the sum is not needed in calculating pY (y1 ) = 0.25.

2.6

Summary and comments

This chapter introduces the axiomatic approach to probability by giving a number of fundamental concepts and results which are at the basis of all further developments in both ﬁelds of probability theory and statistics. In essence, the axiomatic approach consists in calling ‘probability’ any set function that satisﬁes certain properties, together with the deﬁnition of what exactly is meant by the term ‘event’. Clearly, in order to speak of probability this latter deﬁnition is a necessary prerequisite because probabilities can only be assigned to events. Both deﬁnitions, probability and events, are given in Section 2.2 by introducing the concepts of elementary probability spaces – which apply to all cases with a ﬁnite number of possible outcomes – and probability spaces, where the restriction of ﬁniteness is relaxed. These notions are sufﬁciently general to include as special cases all the deﬁnitions of probability considered in Chapter 1. In mathematical terms, a probability space is just a ﬁnite measure space and a probability P is a σ -additive measure deﬁned on a σ -algebra of subsets (the events) of a ‘universal’ set W with the property P(W) = 1. The domain of P is taken to be a σ -algebra because measure theory – by virtue of the construction of the Lebesgue extension of measures – guarantees that a knowledge of the values taken on by P on a limited number of ‘elementary’ events (which, in general, form a semialgebra of subsets of W) is sufﬁcient in order to determine uniquely P on a much broader class of events, this class being, in fact, a σ -algebra. The extension procedure is outlined in Section 2.2.1 and is summarized by the result known as Caratheodory extension theorem. A fundamental aspect of probability which distinguishes it from measure theory is the notion of independent events. Due to its far-reaching consequences in both the theory and real-word applications of statistics, this concept is discussed in some detail in Section 2.2.2, where it is pointed out that the intuitive idea of independence as the absence of causal relation between two (or more) events is translated into mathematical language by a product rule between the probabilities of the events themselves. At this point we consider the fact that in many applications the analyst is mainly interested in assigning probabilities to numerical quantities associated with the outcomes of an experiment rather than to the outcomes themselves. This task is accomplished by introducing the concept of r.v., that is, a real-valued function deﬁned on W and satisfying a ‘measurability’ condition with respect to the σ -algebra of W and the σ -algebra of Borel sets

Probability: the axiomatic approach

73

of the real line R. The condition is formulated by requiring that the inverse image of any Borel set (in the domain of the random variable) be an element of the σ -algebra of W, thus allowing the possibility of assigning probabilities to subsets of R (in the form of open, closed, semiclosed intervals or of individual real numbers, just to mention the most frequently encountered cases). A r.v. X, in turn, induces a probability measure PX on the real line and therefore a real probability space (R, B, PX ). This space, in general, is all that is needed in applications because, through the measure PX , the analyst can obtain a complete probabilistic description of X by deﬁning the so-called PDF of X, usually denoted by FX (x). Clearly, PX and FX are strictly related; in fact, mathematically speaking, PX is the Lebesgue–Stieltjes measure corresponding FX and FX is the generating function the measure PX (this name comes from the fact that in analysis one usually deﬁnes a Lebesgue–Stieltjes measure by means of its generating function and not the other way around as it is done in probability). With the concept of PDF at our disposal, Section 2.3.1 classiﬁes the various types of random variables according to the continuity properties of their PDFs. A ﬁrst classiﬁcation distinguishes between discrete and continuous r.v.s by also introducing the concept of probability mass distribution for discrete r.v.s. Then, among continuous r.v.s a further classiﬁcation distinguishes between absolutely continuous and singular continuous r.v.s, the distinction being due to the fact that for the former type – by far the most important in applications – it is possible to express their PDF by means of the ordinary Lebesgue integral of an appropriate pdf fX (x) which, in turn, is the derivative of FX (x). This possibility relies ultimately on an important result of analysis (given in Appendix B) known as Radon–Nikodym theorem. The conclusion is that the PDF of the most general type of r.v. can be expressed as the sum of a discrete part, an absolutely continuous part and a continuous singular part which, however, is generally absent in most practical cases. Moreover, for the discrete and the absolutely continuous case, respectively, the mass distribution and the pdf provide a complete probabilistic description of the r.v. under study. Proceeding in the discussion of fundamentals, Section 2.3.2 introduces the most common numerical descriptors of r.v.s – the so-called moments of a r.v. – which are deﬁned by means of abstract Lebesgue integrals on the space W. Special cases of moments – the ﬁrst and second moment, respectively – are the familiar quantities known as mean and variance. The properties of moments are then considered together with the important result of Chebychev’s inequality. Subsequently, the problem of actually calculating moments is considered by ﬁrst noting that it is generally not convenient to compute abstract integrals on W. In this regard, in fact, it is shown that moments can be obtained as Lebesgue–Stieltjes integrals on the real line and that these integrals, in the most common cases of discrete and absolutely continuous r.v.s respectively, reduce to a sum and to an ordinary Riemann

74 Probability theory integral. A few examples are then given in order to illustrate some frequently encountered cases. Besides the PDF, another way of obtaining a complete probabilistic description of a r.v. X is its characteristic function ϕX (u). The concept is introduced in Section 2.4 together with some comments on the ‘parallel’ notion of moment generating function (denoted MX (s)). The main properties of CFs are given by also showing how moments (when they exist) can be easily computed by differentiating ϕX . The fact that the CF provides a complete probabilistic description of a r.v. is due to the existence of a one-to-one relationship between PDFs and CFs. The problem of obtaining the CF from the PDF is given by the deﬁnition of CF itself while the reverse problem is addressed by the so-called inversion formulas which, in the general case, are important for their mere existence but are of little practical use. In the particular case of absolutely continuous r.v.s, however, things are easier because the correspondence reduces to the fact that the pdf fX and the CF ϕX are a Fourier transform pair and the notion of Fourier transform is well known and widely used in Engineering and Physics literature. The section closes with a brief discussion of convergence in distribution (or weak convergence) of sequences of r.v.s for its strict relation with pointwise convergence of characteristic functions. Finally, in Section 2.5 we consider a number of complementary ideas and concepts which are worthy of mention in their own right but have been delayed in order not to interrupt the main line of reasoning. Section 2.5.1 introduces the notion of almost-sure (and almost impossible) events by pointing out that there exist events with probability one (and zero) which are different from W (and ∅). This is not surprising in the light of measure theory when, for example, one considers the Lebesgue measure on a ﬁnite interval of the real line. Subsequently (Section 2.5.2) we extend the notion of conditional probability by showing how, in general – given an event with strictly positive probability – there is no difﬁculty in deﬁning such concepts as the conditional PDF of a r.v., the conditional CF, etc. The chapter closes with Section 2.5.3 where it is shown with some examples how to obtain the PDF, pdf or mass distribution of a function g(X) when the PDF (pdf or mass distribution) of the r.v. X is known.

References and further reading [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press,19th printing (1999). [3] Gnedenko B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [4] Haaser, N.B., Sullivan, J.A., ‘Real Analysis’, Dover, New York (1991). [5] Kolmogorov, A.N., ‘Foundations of Probability’, AMS Chelsea Publishing, Providence, Rhode Island (2000).

Probability: the axiomatic approach

75

[6] Kolmogorov, A.N., Fomin, S.V., ‘Introductory Real Analysis’, Dover, New York (1975). [7] Kolmogorov, A.N., Fomin, S.V., ‘Elementi di Teoria delle Funzioni e di Analisi Funzionale’, Edizioni Mir, Mosca (1980). [8] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [9] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000). [10] Rudin, W., ‘Principles of Mathematical Analysis’, 3rd ed., McGraw-Hill, New York, (1976). [11] Rudin, W., ‘Real and Complex Analysis’, McGraw-Hill, New York (1966). [12] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997).

3

3.1

The multivariate case: random vectors

Introduction

The scope of this chapter is to proceed along the line of reasoning of Chapter 2 by turning our attention to cases in which two or more random variables are considered together. With this in mind, we will introduce the new concept of ‘random vector’ by considering measurable vector-valued functions deﬁned on probability spaces. The main mathematical aspects parallel closely the one-dimensional case but it is worth pointing out that now the notion of stochastic independence will play a major role. In fact, this concept is peculiar to probability theory and distinguishes it from being merely an application of analysis.

3.2

Random vectors and their distribution functions

The deﬁnition of random vector is a straightforward generalization of the concept of random variable; in fact Deﬁnition 3.1 Given a probability space (W, S, P), an n-dimensional random vector is a function X: W → Rn such that X−1 (B) ∈ S for every Borel set B ∈ B(Rn ) where, as customary, we denote by B(Rn ) or Bn the σ -algebra of all Borel sets of Rn . In this regard it is important to note that the σ -algebra Bn is the cartesian product of the n terms B × B × · · · × B, meaning, in other words, that every n-dimensional Borel set A ∈ Bn is of the form A = A1 × A2 × · · · × An where A1 , . . . , An are one-dimensional Borel sets. So, in other words, a random vector is a measurable function from W to Rn just as a random variable is a measurable function from W to the real line R. In the present case, however, the vector-valued function X has n components – that is, X = (X1 , X2 , . . . , Xn ) – and the question on the measurability of each individual function Xi (i = 1, . . . , n) arises. The main result is that X is measurable if and only if each function Xi is measurable,

The multivariate case

77

or, equivalently: Proposition 3.1 The vector-valued function X is a random vector if and only if each one of its components Xi is a random variable. As in the one-dimensional case, the original probability space (W, S, P) is of little importance in applications and one should not worry too much about measurability because the concept is sufﬁciently broad to cover almost all cases of practical interest. Therefore, given a random vector X, the analyst’s main concern is the (real) induced probability space (Rn , Bn , PX ), where the probability measure PX is deﬁned by the relation PX (B) ≡ P[X−1 (B)] = P{w ∈ W : X(w) ∈ B}

(3.1a)

for all B ∈ Bn (it is not difﬁcult to show that PX is, indeed, a probability measure). Again, we note that a common ‘shorthand’ notation is to write P(X ∈ B) to mean the probability deﬁned by eq. (3.1a). Also, in the light of the fact that B can be expressed as the cartesian product of n one-dimensional Borel sets B1 , . . . , Bn , the explicit form of eq. (3.1a) is n −1 −1 −1 Xi (Bi ) PX (B) ≡ P[X (B)] = P[X (B1 × · · · × Bn )] = P i=1

(3.1b) By means of PX we can deﬁne the so-called joint probability distribution function (joint-PDF) FX : Rn → [0, 1] as FX (x) = PX {w ∈ W : X1 (w) ≤ x1 , X2 (w) ≤ x2 , . . . , Xn (w) ≤ xn } (3.2a) where x ∈ Rn is the vector whose components are x1 , x2 , . . . , xn and it is understood that all the inequalities on the r.h.s. of eq. (3.2a) must hold simultaneously. In rigorous (and rather cumbersome) notation it may be worth noting that FX (x) can be expressed in terms of the original probability P as n −1 {Xi (−∞, xi ]} FX (x) = P (3.2b) i=1 (and probably this is why, in agreement with the ‘shorthand’ notation above, one often ﬁnds the less intimidating FX (x) = P(X1 ≤ x1 , X2 ≤ x2 , . . . , Xn ≤ xn )). The main properties of the joint-PDF are the natural extensions of (D1)–(D3) given in Chapter 2 and can be summarized as follows: (D1 ) FX (x) = FX (x1 , x2 , . . . , xn ) is non-decreasing and continuous to the right in each variable xi (i = 1, . . . , n),

78 Probability theory (D2 ) limxi →−∞ FX (X) = 0 and limX→∞ FX (x) = 1, where it should be noted that the ﬁrst limit holds for any particular xi tending to −∞ (with all other coordinates ﬁxed) whereas the second property requires that all xi tend to +∞. So, in a different notation, the two properties can be expressed as FX (−∞, x2 , . . . , xn ) = FX (x1 , −∞, . . . , xn ) = · · · = FX (x1 , x2 , . . . , −∞) = 0; FX (+∞, +∞, . . . , +∞) = 1 respectively. In mathematical terminology – as in the one-dimensional case (Section 2.3) – one refers to PX as the Lebesgue–Stieltjes measure determined by FX and, conversely, to FX as the generating function of the (ﬁnite) measure PX . If now we turn our attention to the property expressed by eq. (2.15), we ﬁnd that its multi-dimensional generalization is a bit more involved. For simplicity, let us consider the two-dimensional case ﬁrst. The two-dimensional counterpart of the half-open interval (a, b] is a rectangle R whose points satisfy the inequalities a1 < x1 ≤ b1 and a2 < x2 ≤ b2 ; with this in mind it is not difﬁcult to determine that P(X ∈ R) = FX (b1 , b2 ) − FX (b1 , a2 ) − FX (a1 , b2 ) + FX (a1 , a2 )

(3.3a)

and going over to the more complicated n-dimensional case we get P(X ∈ R) =

(−1)k FX (c1 , c2 , . . . , cn )

(3.3b)

where now (i) R is the n-dimensional parallelepiped (a1 , b1 ] × (a2 , b2 ] × · · · × (an , bn ], (ii) the sum is extended to all the 2n possible choices of the ci ’s being equal to ai or bi – that is, the vertexes of the parallelepiped – and (iii) k represents the number of ci ’s being equal to ai . For instance, if n = 3 we get P(X ∈ R) = FX (b1 , b2 , b3 ) − FX (b1 , b2 , a3 ) − FX (b1 , a2 , b3 ) − FX (a1 , b2 , b3 ) + FX (b1 , a2 , a3 ) − FX (a1 , b2 , a3 ) + FX (a1 , a2 , b3 ) − FX (a1 , a2 , a3 ) So, to every random vector there corresponds a joint-PDF which satisﬁes the properties above. The reverse statement, however, is not true in general unless we add another property to (D1 ) and (D2 ): for a function F to be the joint-PDF of some random vector the sum on the r.h.s. of eq. (3.3b) must be non-negative for any ai , bi such that ai ≤ bi (i = 1, 2, . . . , n). If a function F satisﬁes these three properties, then it is the joint-PDF of some random

The multivariate case

79

vector although, as in the one-dimensional case, this vector is not uniquely determined by F. This is only a minor inconvenience without signiﬁcant consequences in most practical cases. If all the components of a random vector are discrete random variables, we speak of discrete random vector. More speciﬁcally, a random vector X = (X1 , X2 , . . . , Xn ) is discrete if there is a ﬁnite or countable set AX ⊂ Rn such that P[(X1 , X2 , . . . , Xn ) ∈ AX ] = 1; in this case – besides being understood that the set AX is the range of X – the function pX : Rn → [0, 1] deﬁned by pX (x1 , x2 , . . . , xn ) ≡ P(X1 = x1 , X2 = x2 , . . . , Xn = xn )

(3.4)

is called the joint probability mass function (joint-pmf) of X and satisﬁes the normalization condition

pX =

all x

···

all x1

pX (x1 , . . . , xn ) = 1

(3.5)

all xn

The other type of random vector commonly encountered in applications is called jointly absolutely continuous. In this case there is a measurable non-negative function fX on Rn such that for all B ∈ Bn we have PX (B) =

fX dµn

(3.6)

B

where µn denotes here the n-dimensional Lebesgue measure. The function fX (x1 , x2 , . . . , xn ) is called the joint probability density function (joint-pdf) of X and its main properties are the generalization of the one-dimensional case (eqs (2.23), (2.25a) and (2.25b)), that is x1 x2 FX (x1 , x2 , . . . , xn ) =

xn ···

−∞ −∞

fX (t1 , t2 , . . . , tn ) dt1 dt2 · · · dtn

−∞

(3.7a) fX (x1 , . . . , xn ) = ∞ ∞

(3.7b)

∞ ···

−∞ −∞

∂FX (x1 , . . . , xn ) ∂x1 · · · ∂xn

fX (x1 , x2 , . . . , xn ) dx1 dx2 · · · dxn = 1

(3.7c)

−∞

As in the one-dimensional case, discrete and absolutely continuous random vectors, or combinations thereof, are not the only possibilities because Proposition 2.11 on the decomposition of measures still holds in Rn and the

80 Probability theory decomposition of a general PDF, in turn, reﬂects the decomposition of its probability measure. The cases shown above, however, are by far the most common in applications and there is generally no need – besides a speciﬁc theoretical interest – to introduce further complications which, if and whenever necessary, will be considered in future discussions. So, we close this section here and turn our attention, once again, to the important role of stochastic independence. 3.2.1

Marginal distribution functions and independent random variables

In the preceding section we pointed out that each individual component Xi of a random vector X = (X1 , . . . , Xn ) is a random variable itself. Consequently, it becomes important to examine the relation between the joint-PDF of X and the PDF of its components in order to answer the following two questions: (i) given the joint-PDF of X is it possible to determine the PDF of each Xi or the joint-PDF of some of the Xi taken together and forming a random vector with m(m < n) components? (ii) given all the PDFs Fi (x) of each Xi is it possible to obtain the joint-PDF FX (x1 , . . . , xn ) of the random vector X? Let us consider question (i) ﬁrst. The answer is always yes because the joint-PDF of X implicitly contains the joint-PDF of any vector obtained by eliminating some of its components. This PDF can be determined from FX by letting all the components to be eliminated tend to inﬁnity; so, if we call Y the vector obtained by eliminating the kth component of X we have FY (x1 , . . . , xk−1 , xk+1 , . . . , xn ) = lim FX (x1 , . . . , xn ) xk →∞

(3.8)

Similarly, if we eliminate any 2, 3, . . . , n − 1 components of X the PDF of the new vector will be a function of the remaining n − 2, n − 3, . . . , 1 variables, respectively, and the r.h.s. of eq. (3.8) will be a multiple limit where all the variables to be eliminated tend to +∞. All the possible ‘sub-PDFs’, so to speak, obtained like this are called marginal-PDFs of the original vector X. Consider the two-dimensional case as an example; here we have a vector X = (X, Y) whose joint-PDF is the function FX (x, y) (often also denoted by FXY (x, y)) and the two marginal-PDFs are the one-dimensional PDFs of the random variables X and Y, respectively, that is, FX (x) = FXY (x, ∞) ≡ lim FXY (x, y) y→∞

FY (y) = FXY (∞, y) ≡ lim FXY (x, y) x→∞

(3.9)

The multivariate case

81

where, clearly, FX (x) is associated to the probability measure PX : B → [0, 1] and FY (y) is associated to the probability measure PY : B → [0, 1] (we recall that B is the collection of all Borel sets of the real line). So, the ﬁrst part of eq. (3.9) tells us that FX (x) is the probability that the r.v. X takes on a value less than or equal to x when all the possible values of Y have been taken into account and, similarly, the second part of eq. (3.9) states that FY (y) is the probability that the r.v. Y takes on a value less than or equal to y when all the possible values of X have been taken into account. Therefore – continuing with the two-dimensional case for simplicity – it is not difﬁcult to see that the marginal-pmfs of a discrete random vector are given by pX (x) =

pXY (x, y)

all y

pY (y) =

(3.10a)

pXY (x, y)

all x

where pXY (x, y) is the joint-pmf of the two r.v.s X, Y forming the vector X. Similarly, if X is a two-dimensional absolutely continuous random vector with joint-pdf fXY (x, y) = ∂FXY /∂x∂y, the two marginal (one-dimensional) pdfs are fX (x) =

+∞ fXY (x, y) dy −∞ +∞

fY (y) =

(3.10b)

fXY (x, y)dx −∞

Example 3.1(a) joint-pmf is

Let X be a discrete two-dimensional random vector whose

pXY (x, y) = (1 − q)2 q x+y

(3.11)

where q is a constant 0 ≤ q < 1 and the variables x, y can only take on natural values (i.e. 0, 1, 2, . . .). Then eq. (3.10a) yields for the marginal-pmfs pX (x) = q x (1 − q)2 pY (y) = qy (1 − q)2

qy = (1 − q)q x

all y q x = (1 − q)qy

(3.12)

all x

because the series n qn converges to (1 − q)−1 whenever 0 ≤ q < 1. In addition, the reader is invited to verify that the joint-pmf (3.11) satisﬁes the

82 Probability theory normalization condition of eq. (3.5) which, in this case, is written

(1 − q)2 q x+y = 1

all x all y

Example 3.1(b) Let the two-dimensional absolutely continuous random vector X have the joint-pdf fX (x, y) =

( ' 1 1 √ exp − (x2 + xy + y2 ) 3 2π 3

(3.13a)

then, using the ﬁrst of eqs (3.10b) we can obtain the marginal-pdf of the r.v. X by integrating (3.13a) in dy over the entire real line. In order to do so we start by rewriting fX (x, y) as

1 x 2 1 x2 fX (x, y) = exp − y+ √ exp − 4 3 2 2π 3

(3.13b)

so that the ﬁrst exponential can be factored out of the integral in dy. Then, by performing the change of variable t = y + x/2 we get 1 fX (x) = √ exp(−x2 /4) 2π 3

∞

−∞

exp(−t 2 /3) dt (3.14a)

1 = √ exp(−x2 /4) 2 π ∞ √ where in the last equality we used the result −∞ exp(−ax2 ) dx = π/a (which can easily be found in integral tables). Finally, by symmetry, it is immediate to obtain 1 fY (y) = √ exp(−y2 /4) 2 π

(3.14b)

If now we consider question (ii) posed at the beginning of this section it turns out that its answer, in the general case, is no. More speciﬁcally, one cannot determine the joint-PDF of the random vector X = (X1 , . . . , Xn ) from the PDFs of its components Xi unless they are independent. In mathematical terms the following proposition holds Proposition 3.2(a) Let X1 , . . . , Xn be random variables on the probability space (W, S, P), let Fi (xi ) be the PDF of Xi (i = 1, . . . , n) and FX (x1 , . . . , xn )

The multivariate case

83

be the joint-PDF of the vector X = (X1 , . . . , Xn ). Then X1 , . . . , Xn are independent if and only if FX (x1 , . . . , xn ) = F1 (x1 )F2 (x2 ) . . . Fn (xn )

(3.15)

for all real x1 , . . . , xn . It should be noted that Proposition 3.2(a) is an ‘if and only if’ statement; this means that if X1 , . . . , Xn are independent then their joint-PDF can be obtained by taking the product of the individual PDFs and, conversely, if the joint-PDF of a random vector X is the product of n one-dimensional PDFs, then the components X1 , . . . , Xn are independent random variables. At this point, however, we must take a step back and return to the notion of stochastic independence introduced in Chapter 2. In Section 2.2.2, in fact, we discussed in some detail the notion of stochastic independence of events but nothing has been said on independent random variables; we do it now by giving the following deﬁnition Deﬁnition 3.2 A collection of random variables X1 , X2 , . . . is called an independent collection if for any arbitrarily chosen class of Borel sets B1 , B2 , . . . the events X1−1 (B1 ), X2−1 (B2 ), . . . are collectively independent. This deﬁnition means that the product rule (2.9) must apply. So, in particular, n random variables X1 , . . . , Xn are called independent if for any choice of Borel sets B1 , . . . , Bn we have ⎛ P⎝

n

k=1

⎞ Xk−1 (Bk )⎠ =

n

P[Xk−1 (Bk )]

k=1

which, in turn, implies that PX (B1 × B2 × · · · × Bn ) = PX1 (B1 )PX2 (B2 ) · · · PXn (Bn ). We have the following result: Proposition 3.2(b) Let X1 , . . . , Xn be random variables on the probability space (W, S, P), then they are independent if and only if the measure PX is the product of the n individual PXi (i = 1, 2, . . . , n). In the light of the fact that the individual PDFs Fi (xi ) are deﬁned by the probabilities PXi , it is not surprising that Proposition 3.2(a), as a matter of fact, is a consequence of Proposition 3.2(b). Also – as it has been done for events – one can introduce the concept of ‘collection of pairwise independent random variables’ – that is, a set of r.v.s X1 , X2 , . . . where Xi is independent of Xj for each pair of distinct indexes i, j – and note that pairwise independence does not imply independence. The converse, however, is true and it is

84 Probability theory evident that these two statements parallel closely the remarks of Chapter 2 (Section 2.2.2). For discrete and absolutely continuous random variables independence can be characterized in terms of pmfs and pdfs, respectively, because the product rule applies to these functions. More speciﬁcally, if X1 , . . . , Xn are a set of independent random variables on a probability space (W, S, P) then, with obvious meaning of the symbols, pX (x1 , . . . , xn ) = p1 (x1 )p2 (x2 ) · · · pn (xn )

(3.16a)

in the discrete case and fX (x1 , . . . , xn ) = f1 (x1 )f2 (x2 ) · · · fn (xn )

(3.16b)

in the absolutely continuous case. Conversely, if – as appropriate – eq. (3.16a) or (3.16b) applies then the random variables X1 , . . . , Xn are independent. In this regard, for example, it may be worth noting that the two random variables X, Y of Example 3.1(a) are independent because (see eqs (3.11) and (3.12)) pXY (x, y) = pX (x)pY (y). On the other hand, the random variables X, Y of Example 3.1(b) are not independent; in fact, in this case the joint-pdf fX (x, y) cannot be factored as required in eq. (3.16b) because of the cross-term xy in the exponential. One word of caution on the absolutely continuous case is in order: if the random vector X has a pdf fX then each Xi has a pdf fi and eq. (3.16b) holds if X1 , . . . , Xn are independent. However, from the fact that each Xi has a density it does not necessarily follow that X = (X1 , . . . , Xn ) has a density; it does if X1 , . . . , Xn are independent and this density fX is given – as shown by eq. (3.16b) – by the product of the n pdfs fi . As a ﬁnal remark for this section we point out an important property of independent random variables: measurable functions of independent r.v.s are independent r.v.s. More speciﬁcally, we can state the following result whose proof, using the deﬁnition of independence of the Xi , is almost immediate. Proposition 3.3 Let X1 , . . . , Xn be a set of independent random variables and g1 , . . . , gn a set of Borel functions. Then the random variables Z1 , . . . , Zn (we recall from Chapter 2 that Borel functions of r.v.s are r.v.s themselves) deﬁned by the relations Zi ≡ gi (Xi ) (i = 1, 2, . . . , n) are independent. More generally, if Y1 , . . . , Ym are sub-vectors of a vector X such that none of the components of X is a component of more that one of the Yj and g1 , . . . , gm are measurable functions, then Zi ≡ gi (Yi )(i = 1, 2, . . . , m) are independent. Also, one can proceed further. In fact, it is possible to extend Deﬁnition 3.1 in order to deﬁne the independence of n random vectors X1 , . . . , Xn and determine that also in this case the factorization property of the PDFs is a necessary and sufﬁcient condition for independence. For the

The multivariate case

85

moment, however, the results given here will sufﬁce and we delay further considerations on independence to later sections.

3.3

Moments and characteristic functions of random vectors

For simplicity, let us ﬁrst consider a two-dimensional random vector X = (X, Y) deﬁned on a probability space (W, S, P). This is a frequently encountered case in applications and it is worthy of consideration in its own right before generalizing to higher dimensional vectors. As in the one-dimensional case (Section 2.3.2), the moments are deﬁned as abstract Lebesgue integrals in dP. So, if i, j are two non-negative integers the joint-moments of order i + j – denoted by E(X i Y j ) or mij – are deﬁned as mij = E(X i Y j ) =

X i Y j dP

(3.17a)

W

which, in the absolutely continuous case, becomes ∞ ∞ xi yj f (x, y) dx dy

mij =

(3.17b)

−∞ −∞

and the integrals are replaced by the appropriate sums in the discrete case. In the light of eq. (3.10b) and their discrete counterparts (3.10a), it is then clear that the ﬁrst-order moments m10 and m01 , respectively, are simply µX = E(X) and µY = E(Y), that is, the mean values of the individual random variables X and Y and, similarly, mi0 and m0j are the ith moment of X and the jth moment of Y. The central (joint) moments of order i +j, in turn, are deﬁned as (provided that µX , µY < ∞) µij = E[(X − µX )i (Y − µY )j ]

(3.18)

where, in the important case i + j = 2 (second-order central moments) we have µ20 = σX2 = Var(X) and µ02 = σY2 = Var(Y). The moment µ11 is given a special name and is called the covariance of the two variables X, Y. For this reason µ11 is often denoted by Cov(X, Y) – although the symbols XY , σXY and KXY are also frequently found in literature. Besides the immediate relations Cov(X, X) = σX2 , Cov(Y, Y) = σY2 and Cov(X, Y) = Cov(Y, X) it should also be noted that the notion of covariance was mentioned in passing

86 Probability theory in Proposition 2.15 (Section 2.3.2) where, in addition, it was shown that Cov(X, Y) = E(XY) − E(X)E(Y) = m11 − µX µY

(3.19a)

This equation is, broadly speaking, the ‘mixed-variables’ counterpart of eq. (2.34), which we rewrite here for the two individual r.v.s X, Y σX2 = µ20 = E(X 2 ) − E2 (X) = m20 − µ2X σY2 = µ02 = E(Y 2 ) − E2 (Y) = m02 − µ2Y

(3.19b)

Some of the main properties of the abstract integral can be immediately be re-expressed in terms of moments. We have already considered linearity for n random variables (Proposition 2.13(c)) – which, in our present case of two r.v.s reads E(aX + bY) = aE(X) + bE(Y) where a, b are any two real or complex constants – but, in particular, we want to point out here the following inequalities (a) Holder’s inequality: let p, q be two numbers such that p > 1, q > 1 and 1/p + 1/q = 1, then E(|XY|) ≤ [E(|X|p )]1/p [E(|Y|q )]1/q

(3.20)

(b) Cauchy–Schwarz inequality ) ) 2 E(|XY|) ≤ E(X ) E(Y 2 )

(3.21)

Both relations are well known to the reader who is familiar with the theory of Lebesgue-integrable function spaces and, clearly, eq. (3.21) is a special case of (3.20) when p = q = 2. In particular – since the Cauchy–Schwarz inequality holds for any two r.v.s – there is no loss of generality in considering the two centered r.v.s W = X − µX , Z = Y − µY and rewriting (3.21) as E(WZ) = Cov(XY) ≤ σW σZ = σX σY , where the last equality holds because the variance of a constant is zero. Therefore, if one deﬁnes the correlation coefﬁcient ρXY as ρXY =

Cov(XY) σX σY

(3.22)

it is immediate to determine that −1 ≤ ρXY ≤ 1. The fact that whenever ρXY = −1 or ρXY = 1 there is a perfect linear relationship between the two r.v.s X and Y (i.e. a relation of the type Y = aX + b, where a, b are two constants) is not so immediate and requires some explanation. In order to do this as an exercise, note that both equalities ρXY = ±1 imply

The multivariate case

87

E2 (WZ)/E(W 2 )E(Z2 )

= 1, where W, Z are the two centered r.v.s deﬁned above. This relation can be rewritten as −

E2 (WZ) = −E(Z2 ) or, equivalently E(W 2 )

E2 (WZ) E(WZ) E(W 2 ) − 2 E(WZ) + E(Z2 ) = 0 E2 (W 2 ) E(W 2 ) so that setting E(WZ)/E(W 2 ) = a we get a2 E(W 2 )−2aE(WZ)+E(Z2 ) = 0, that is E[(aW − Z)2 ] = 0. This, in turn, means that aW − Z = const, that is, that W and Z are linearly related. Then, since (by deﬁnition) there is a linear relation between W and X and between Z and Y, it follows that also X and Y must be linearly related. On the other hand, in order to prove that Y = aX + b implies ρXY = 1 or ρXY = −1 (depending on whether a > 0 or a < 0) it is sufﬁcient to note that in this case Cov(XY) = aσX and σY = |a|σX ; consequently, ρXY = ±1 follows from the deﬁnition of correlation coefﬁcient. The opposite extreme to maximum correlation occurs when ρXY = 0, that is, when Cov(XY) = 0 (if, as always implicitly assumed here, both σX , σY are ﬁnite and different from zero). In this case we say that X and Y are uncorrelated and then, owing to eq. (3.19a), we get E(XY) = E(X)E(Y). This form of ‘multiplication rule’ for expected values may suggest independence of the two random variables because the following proposition holds: Proposition 3.4

If X, Y are two stochastically independent r.v.s then

E(XY) = E(X)E(Y)

(3.23)

and therefore Cov(XY) = 0. This result can be proven by using the factorization properties given in eqs (3.15), (316a) and (3.16b), but the point here is that the reverse statement of Proposition 3.4 is not, in general, true (unless in special cases which will be considered in future sections). In fact, it turns out that uncorrelation – that is, Cov(XY) = 0 – is a necessary but not sufﬁcient condition for stochastic independence. In other words, two uncorrelated r.v.s are not necessarily unrelated (a term which, broadly speaking, is a synonym of independent) because uncorrelation implies a lack of linear relation between them but not necessarily a lack of relation in general. The following example illustrates this situation. Example 3.2 Consider a random vector (X, Y) which is uniformly distributed within a circle of radius r centered about the origin. This means

88 Probability theory that the vector is absolutely continuous with joint-pdf given by 1/π r2 , x2 + y2 < r2 fXY (x, y) = 0, otherwise so that, for instance, if X = 0 then Y can have any value between −r and r but if X = r then Y can only be zero. Therefore, since knowledge of X provides some information on Y, the two variables are not independent. On the other hand, they are uncorrelated because the symmetry of the problem leads to the result Cov(XY) = 0. In fact, denoting by C the domain where fXY = 0, we can calculate the covariance as (see eq. (3.44)) Cov(XY) =

xy f (x, y) dx dy = C

1 π r2

xy dx dy C

and all the integrals in the four quadrants have the same absolute value. However, the function xy under the integral sign is positive in the ﬁrst and third quadrant and negative in the second and fourth quadrant so that summing all the four contributions yields Cov(XY) = 0. As a simpler example consider a r.v. X with the following characteristics: (a) its pdf (or pmf if it is discrete) is symmetrical about the ordinate axis and (b) it has a ﬁnite fourth moment. Then, if we deﬁne the r.v. Y = X 2 it is immediate to determine that X and Y are uncorrelated but not independent. The deﬁnition of characteristic function (or, more precisely, joint-CF) for a two-dimensional random vector is a simple extension of the one-dimensional case of Section 2.4 and we have φ(u, v) = E[ei(uX+vY) ]

(3.24a)

which, in view of generalization to higher dimensions, can be expressed more synthetically with the aid of matrix algebra. We denote by u the vector whose components are the two real variables u, v and write ϕX (u) = E[exp(iuT X)]

(3.24b)

where, following the usual matrix notation, two-dimensional vectors are expressed as column matrices and their transpose (indicated by the upper T) are therefore row matrices. So, in eq. (3.24b) it is understood that the matrix

The multivariate case

89

multiplication in the exponential reads X uT X = (u, v) = uX + vY Y The properties of the CF of a random vector parallel closely the onedimensional case; in particular ϕX (u) is uniformly continuous on R2 and, in addition ϕX (0) = 1 |ϕX (u)| ≤ 1 for all u ∈ R2 ∗ ϕX (−u) = ϕX (u)

(3.25)

where 0 = (0, 0) is the zero vector and the asterisk denotes complex conjugation. Also, if we set n = j + k (where j, k are two integers) and the vector X has ﬁnite moments of order n, then ϕX (u, v) is j times derivable with respect to u and k times derivable with respect to v and ∂ n ϕX (u, v) n n j k i mjk = i E(X Y ) = (3.26) ∂ j u∂ k v u=v=0 which is the two-dimensional counterpart of eq. (2.47a). Equation (3.26) shows that the moments of a random vector coincide – besides the multiplicative factor 1/in – with the coefﬁcients of the MacLaurin expansion of ϕX . This implies that the existence of all moments allows one to construct the MacLaurin series of the CF although, as in the one-dimensional case, in general it does not allow to reconstruct ϕX itself. The marginal CFs of any ‘sub-vector’ of X can be obtained from ϕX by simply setting to zero all the arguments corresponding to the random variable(s) which do not belong to the sub-vector in question. This is an immediate consequence of the deﬁnition of CF and in the two-dimensional case under study we have ϕX (u) = ϕX (u, 0) ϕY (v) = ϕX (0, v)

(3.27)

As ﬁnal remarks to this section, two results are worthy of notice. The ﬁrst is somehow expected and states: Proposition 3.5 Two random variables X, Y forming a vector X are stochastically independent if and only if ϕX (u, v) = ϕX (u)ϕY (v)

(3.28)

meaning that the product rule (3.28) is a necessary and sufﬁcient condition for independence.

90 Probability theory A word of caution is in order here because Proposition 3.5 should not be confused with a different result (see also Example 3.3) which states that if X, Y are independent, then ϕX+Y (u) = ϕX (u)ϕY (u)

(3.29)

In fact, the independence of X and Y imply the independence of the r.v.s ei uX and ei uY and consequently E(ei u (X+Y) ) = E(ei uX ei uY ) = E(ei uX )E(ei uY ) by virtue of Proposition 3.4. Then, by the deﬁnition of CF we get E(ei uX )E(ei uY ) = ϕX (u)ϕY (u). The converse of this result, however, is not true in general and the equality (3.29) does not imply the independence of X and Y. These same considerations, clearly, can be extended to the case of more than two r.v.s. The second remark – here already given in the n-dimensional case – has to do with the important fact that the joint-CF ϕX provides a complete probabilistic description of the random vector X = (X1 , . . . , Xn ) because the joint-PDF FX (x1 , . . . , xn ) is uniquely determined by ϕX (u1 , . . . , un ). The explicit result, which we state here for completeness, is in fact the n-dimensional counterpart of Proposition 2.20 and is expressed by the relation 1 P(ak < Xk ≤ bk ) = lim c→∞ (2π )n ×

n

k=1

c

c ···

−c

−c

eiuk ak − eiuk bk iuk

ϕX (u1 , . . . , un ) du1 · · · dun (3.30)

where the real numbers ak , bk (k = 1, . . . , n) delimitate a bounded parallelepiped (i.e. an interval in Rn ) whose boundary has zero probability measure. 3.3.1

3.3.1.1

Additional remarks: the multi-dimensional case and the practical calculation of moments The multi-dimensional case

In the preceding section we have been mainly concerned with twodimensional random vectors but it is reasonable to expect that most of the considerations can be readily extended to the n-dimensional case. We only outline this extension here because it will not be difﬁcult for the reader to ﬁll in the missing details. It is implicitly assumed, however, that the reader has some familiarity with matrix notation and basic matrix properties (if

The multivariate case

91

not, one may refer, for example, to Chapter 11 of Ref. [3] at the end of this chapter, to the excellent booklet [16] or to the more advanced text [8]). Given a n-dimensional random vector X = (X1 , . . . , Xn ) and a positive integer k, the kth order moments are deﬁned as m(k1 , k2 , . . . , kn ) = E X1k1 X2k2 . . . Xnkn

(3.31)

where k1 , . . . , kn are n non-negative integers such that k = k1 +k2 +· · ·+kn . This implies that we have now n ﬁrst-order moments – which can be denoted m1 , m2 , . . . , mn – and n2 second-order ordinary and central moments. These latter quantities are often conveniently arranged in the so-called covariance matrix ⎛

K11 ⎜K21 ⎜ K=⎜ . ⎝ ..

K12 K22 .. .

Kn1

Kn2

⎞ . . . K1n . . . K2n ⎟ ⎟ T .. ⎟ = E[(X − m)(X − m) ] .. . ⎠ . . . . Knn

(3.32a)

where Kij = Cov(Xi Xj ) with i, j = 1, . . . , n and in the second expression (X−m) is the n×1 column matrix whose elements are X1 −m1 , . . . , Xn −mn , that is, the difference of the two column matrices X = (X1 , . . . , Xn )T and the ﬁrst-order moments matrix m = (m1 , . . . , mn )T . In this light, it is easy to notice that the covariance matrix can be written as K = E(XXT ) − mmT

(3.32b)

The matrix K is obviously symmetric (i.e. Kij = Kji or, in matrix symbolism, K = KT ) so that there are only n(n + 1)/2 distinct elements; also it is evident that the elements on the main diagonal are the variances of the individual r.v.s – that is, Kii = Var(Xi ). Similar considerations of symmetry and of number of distinct elements apply to the correlation matrix R deﬁned as ⎛

1 ⎜ρ21 ⎜ R=⎜ . ⎝ ..

ρ12 1 .. .

ρn1

ρn2

⎞ . . . ρ1n . . . ρ2n ⎟ ⎟ .. ⎟ .. . ⎠ . ... 1

(3.32c)

where (eq. (3.22)) ρij = Kij /σi σj and, for brevity, we denote by σi = √ Var(Xi ) the standard deviation of the r.v. Xi (assuming that σi is ﬁnite for each i = 1, . . . , n). The relation between K and R – as it is immediately

92 Probability theory veriﬁed by using the rules of matrix multiplication – is K = SRS

(3.33)

where we called S = diag(σ1 , σ2 , . . . , σn ) the matrix whose the only nonzero elements are σ1 , . . . , σn on the main diagonal. If the n r.v.s are pairwise uncorrelated – or, which is a stronger condition, pairwise independent – then both K and R are diagonal matrices and in particular R = I, where I is the identity, or unit, matrix (its only non-zero elements are ones on the main diagonal). Clearly, this holds true if the r.v.s Xi are mutually independent; in this case we can also generalize Proposition 3.4 on ﬁrst-order moments to the multiplication rule n n

Xi = E(Xi ) (3.34) E i=1

i=1

If we pass from the random vector X to a m-dimensional random vector Y by means of a linear transformation Y = AX – where A is a m × n matrix of real numbers – we can use the second expression of (3.32a) to determine Y the covariance matrix KY of Y in terms of KX . In fact, calling for brevity the ‘centered’ matrices Y − mY and X − mX , respectively, we get and X T] Y YT ) = E[(AX)(A X) KY = E( X T AT ] = A E(X X T )AT = A KX AT = E[AX

(3.35)

where we used the well-known relation stating that the transpose of a product of matrices equals the product of the transposed matrices taken in reverse T AT . We mention here in passing a T =X order – that is, in our case (AX) ﬁnal property of the covariance and correlation matrices: both K and R are positive semi-deﬁnite. As it is known from matrix theory, this means that zT Kz ≥ 0

(3.36)

where z is a column vector of n real or complex variables and xT Kx is the so-called quadratic form of the (symmetric) matrix K. Equation (3.36) implies that det(K) – that is, the determinant of K, often also denoted by |K| – is non-negative. Clearly, the same considerations apply to R. The characteristic function of a n-dimensional random vector is the straightforward extension of eq. (3.24b) and the generalization of eq. (3.26) reads k ϕ (u) ∂ X (3.37) ik m (k1 , . . . , kn ) = k1 kn ∂u · · · ∂un 1

u=0

provided that the moment of order k = k1 + k2 + · · · + kn exists.

The multivariate case

93

The marginal CFs can be obtained from ϕX (u1 , . . . , un ) as stated in Section 3.3. For example, ϕX (u1 , u2 , . . . , un−1 , 0) is the joint-CF of the vector (X1 , . . . , Xn−1 ), ϕX (u1 , 0, . . . , 0) is the one-dimensional CF of the r.v. X1 , etc., and the multiplication rule ϕX (u1 , . . . , un ) =

n

ϕXi (ui )

(3.38)

i=1

is a necessary and sufﬁcient condition for the mutual independence of the r.v.s Xi (i = 1, . . . , n). Similarly, all the other considerations apply. In addition, we can determine how a joint-CF changes under a linear transformation from X to a m-dimensional random vector Y. As above, the transformation is expressed in matrix form as Y = AX and we assume here that ϕX (u) is known so that ϕY (v) = E[eiv

T

Y

= E[ei(A

T

] = E[eiv T

v) X

T

AX

]

] = ϕX (AT v)

(3.39a)

In the more general case Y = AX + b – where b = (b1 , . . . , bm )T is a column vector of constants – then it is immediate to determine T

ϕY (v) = ei v b ϕX (AT v)

(3.39b)

In the preceding section nothing has been said about moment-generating functions (MGFs) but by now it should be clear that the deﬁnition is MX (s1 , . . . , sn ) = E[exp(sT X)]

(3.40)

where s1 , . . . , sn is a set of n variables. Within the limitations on the existence of MX outlined in Section 2.4 we have the n-dimensional version of eq. (2.48), that is, ∂ k MX (s) m(k1 , . . . , kn ) = k (3.41) ∂s 1 · · · ∂sknn 1

3.3.1.2

s=0

The practical calculation of expectations

Many quantities introduced so far – moments in the ﬁrst place but CFs and MGFs as well – are deﬁned as expectations, which means, by deﬁnition, as abstract Lebesgue integral in dP. Therefore, the problem arises of how these integrals can be calculated in practice. With the additional slight complication of n-dimensionality, the general line of reasoning parallels closely all that has been said in Chapter 2. We will brieﬂy repeat it here.

94 Probability theory Given a random vector X on the probability space (W, S, P) – that is, a measurable function X : W → Rn – we can make probability statements regarding any Borel set B ∈ Bn by considering the probability measure PX deﬁned by eq. (3.1) and working in the induced real probability space (Rn , Bn , PX ). This is the space of interest in practice, PX being a Lebesgue– Stieltjes measure (on Rn ) which, by virtue of eq. (3.2), can be associated with the PDF FX . At this point we note that the n-dimensional version of Proposition 2.17 applies, with the consequence that our abstract Lebesgue integral on W can be calculated as a Lebesgue–Stieltjes integral on Rn . Then, depending on the type of random vector under study – or, equivalently, on the continuity properties of FX – this integral turns into a form amenable to actual calculations. As in the one-dimensional case, there are three possible cases: the discrete, the absolutely continuous and the singular continuous case, the ﬁrst two (or a mixture thereof) being by far the most important in practice. If X is a discrete random vector its range is a discrete subset AX ⊂ Rn and its complete probabilistic description can be given in terms of the mass distribution pX (x) (see also eq. (3.4)) which, in essence, is a ﬁnite or countable set of real non-negative numbers pi1 ,i2 ,...,in (the n indexes i1 , . . . , in mean that the ith r.v. Xi can take on the values xi1 , xi2 , . . .) such that the normalization condition

pX (x) =

pi1 ,...,in = 1

(3.42)

i1 ,...,in

x∈AX

holds. In this light, given a Borel measurable function g(x) its expectation is given by the sum E[g(x)] =

g(x)pX (x)

(3.43)

x∈AX

Also, the marginal mass distribution of any group of any m(m < n) random variables is obtained by summing the pi1 ,i2 ,...,in over all the n − m remaining variables; so, for example, the marginal mass distribution of the vector (X1 , . . . , Xn−1 ) is given by in pi1 ,i2 ,...,in . This is just a straightforward generalization of eq. (3.10a). If X is absolutely continuous there exists a density function fX (x) such that eqs (3.7a–3.7c) hold. Then, the expectation of a measurable function g(x) becomes a Lebesgue integral and reads E[g(x)] =

g(x)fX (x) dx Rn

(3.44)

The multivariate case

95

Rn .

where dx is the Lebesgue measure on As one might expect – since the Lebesgue integral is, broadly speaking, a generalization of the ordinary Riemann integral of basic calculus – the integral (3.44) coincides with Riemann’s (when this integral exists). In the multi-dimensional case, however, a result of fundamental importance is Fubini’s theorem (Appendix B) which guarantees that a Lebesgue multiple integral can be calculated as an iterated integral under rather mild conditions on the integrand function. This theorem is the key to the practical evaluation of multi-dimensional integrals. Two ﬁnal comments are in order before closing this section. First, we recall eq. (3.10b) and note that their n-dimensional extension is immediate; in fact, the marginal pdfs of any ‘subvector’ of X of m(m < n) components is obtained by integrating fX with respect to the remaining n − m variables. So, for instance, fX1 (x1 ) = Rn−1 fX (x1 , . . . , xn ) dx2 · · · dxn is the pdf of ∞ the r.v. X1 and fY (x1 , . . . , xn−1 ) = −∞ fX (x1 , . . . , xn ) dxn is the (n − 1)dimensional joint-pdf of the random vector Y = (X1 , X2 , . . . , Xn−1 ). The second comment has to do with the CF of an absolutely continuous random vector X. Owing to eq. (3.44), in fact, ϕX (u) and fX (x) turn out to be a Fourier transform pair so that we have fX (x) ei u

ϕX (u) =

T

x

dx

(3.45a)

Rn

with the inversion formula 1 fX (x) = (2π )n

3.3.2

ϕX (u) e−i u

T

x

du

(3.45b)

Rn

Two important examples: the multinomial distribution and the multivariate Gaussian distribution

A multinomial trial with parameters p1 , p2 , . . . pn is a trial with n possible outcomes where the probability of the ith outcome is pi (i = 1, 2, . . . , n) and, clearly, p1 + p2 + · · · + pn = 1. If we perform an experiment consisting of N independent and identical multinomial trials (so that the pi do not change from trial to trial) we may call Xi the number of trials that result in outcome i so that each Xi is a r.v. which can take on any integer value between zero and N. Forming the vector X = (X1 , . . . , Xn ), its joint-pmf is called multinomial (it can be obtained with the aid of eq. (1.16)) and we have p(x1 , . . . , xn ) =

N! px1 , px2 · · · pxnn x1 !x2 ! . . . xn ! 1 2

(3.46a)

96 Probability theory where, since each xi represents the number of times in which i occurs n

xi = N

(3.46b)

i=1

The name multinomial is due to fact that the expression on the r.h.s. of (3.46a) is the general term in the expansion of (p1 + · · · + pn )N . If n = 2 eq. (3.46a) reduces to the binomial pmf considered in Examples 2.8 and 2.9 (a word of caution on notation: in eq. (2.41a) n is the total number of trials while here n is the number of possible outcomes in each trial). As an easy example we can consider three throws of a fair die. In this case N = 3 and n = 6, xi is the number of times the face 1 shows up, x2 is the number of times the face 2 shows up, etc. and p1 = · · · = p6 = 1/6. As a second example we can think of a box with, say, 50 balls of which 10 are white, 22 yellow and 18 are red. The experiment may consist in extracting – with replacement – 5 balls from the box and then counting the extracted balls of each color. In this case N = 5, n = 3 and p1 = 0.20, p2 = 0.44, p3 = 0.36. Note that after each extraction it is important to replace the ball in the box, otherwise the probabilities pi would change from trial to trial and one of the basic assumptions leading to (3.46) would fail. The joint-CF of the multinomial distribution is obtained from eq. (3.24b) by noting that in this discrete case the Lebesgue–Stieltjes integral deﬁning the expectation becomes a sum on all the xi s. Therefore

N! px1 · · · pxnn ei(u1 x1 +···+un xn ) x1 ! . . . xn ! 1 N! (p1 eiu1 )x1 · · · (pn eiun )xn = x1 ! . . . xn !

ϕX (u1 , . . . , un ) =

(3.47)

= (p1 eiu1 + · · · + pn eiun )N As a second step, let us obtain now the marginal CF of one of the r.v.s Xi , for example, X1 . In order to do this (recall Sections 3.2 and 3.3) we must set u2 = u3 = · · · = un = 0 in eq. (3.47) thus obtaining ϕX1 (u1 ) = (p1 eiu1 + p2 + · · · + pn )N = (1 − p1 + p1 ei u1 )N

(3.48)

which is the CF of a one-dimensional binomial r.v. (eq. (2.51)). Also, using the ﬁrst of (2.47b) we can obtain the ﬁrst moment of X1 , that is, E(X1 ) = Np1

(3.49)

in agreement with eq. (2.41b) and with the result we would get by using eq. (3.37a) and calculating the derivative ∂ϕX (u)/∂u1 |u=0 of the joint-CF

The multivariate case

97

(3.47). Clearly, by substituting the appropriate index, both eqs (3.48) and (3.49) apply to each one of the Xi . At this point one may ask about the calculation of E(X1 ) by directly using eq. (3.43) without going through the CF. This calculation is rather cumbersome but we outline it here for the interested reader. We have E(X1 ) =

all xi

N x1 x1 N! x1 px11 · · · pxnn = N(1 − p1 )N−1 p x1 ! . . . xn ! x1 ! 1 x1 =0

= Np1 (1 − p1 )N−1

N x1 =1

(3.50)

x1 x1 −1 p x1 ! 1

where we ﬁrst isolated the sum over x1 , then used the multinomial theorem for the indexes 2, 3, . . . , N and took into account that p2 + p3 + · · · + pn = 1 − p1 . Then, starting from the multinomial theorem all xi

N! px1 · · · pxnn = (p1 + p2 + · · · + pn )N x1 ! · · · xn ! 1

we can differentiate both sides with respect to p1 and then, on the l.h.s. of the resulting relation, isolate the sum on x1 . This procedure leads in the end to (1 − p1 )N−1

N x1 x1 −1 p =1 x1 ! 1

x1 =1

which, in turn, can be substituted in the last expression of (3.50) to give the desired result E(X1 ) = Np1 . After this, it is evident that the shortest way to determine the covariance between any two r.v.s Xk , Xm (where k, m are two integers < n with k = m) is by using the CF. If we recall that Cov(Xk Xm ) = E(Xk Xm ) − E(Xk )E(Xm ) then we only need to calculate the ﬁrst term on the r.h.s. because, owing to (3.49), the second term is N 2 pk pm . Performing the prescribed calculations we get ∂ 2 ϕX (u) E(Xk Xm ) = − = N(N − 1)pk pm (3.51) ∂uk ∂um u=0

and therefore the off-diagonal terms of the covariance matrix K are given by Cov(Xk Xm ) = −Npk pm

(3.52)

By the same token, for any index 1 ≤ k ≤ n, it is not difﬁcult to obtain E(Xk2 ) = Npk [(N − 1)pk + 1] so that the elements on the main diagonal

98 Probability theory of K are Var(Xk ) = Npk (1 − pk )

(3.53)

At ﬁrst sight – since we spoke of independent trials – the fact that the variables Xi are correlated may seem a bit surprising. The correlation is due to the ‘constraint’ eq. (3.46b) and the covariances are negative (eq. (3.52)) because an increase of any one xi tends to decrease the others. The fact that there exists one constraint equation on the xi implies that K is singular (i.e. det(K) = 0) and has rank n − 1 so that, in essence, the n-dimensional vector X belongs to the (n − 1)-dimensional Euclidean space. Let us consider now the multi-dimensional extension of the Gaussian (or normal) probability law considered in Examples 2.4, 2.8 and 2.9(b). For simplicity, we begin with the two-dimensional case. In the light of eq. (3.16b), the joint-pdf of two independent and individually normal r.v.s X, Y forming a vector X must be # " 1 1 (x − m1 )2 (y − m2 )2 fX (x, y) = (3.54) exp − + 2π σ1 σ2 2 σ12 σ22 where m1 = E(X), σ12 = Var(X) and m2 = E(Y), σ22 = Var(Y). Also, using the result of eqs (2.52) and (3.28) of Proposition 3.5, the joint-CF of X is * + 1 2 2 2 2 ϕX (u, v) = exp i(um1 + vm2 ) − (σ1 u + σ2 v ) 2

(3.55)

which is easy to cast in matrix form as

1 ϕX (u) = exp iu m − uT Ku 2

T

(3.56)

where, in the present case, it should be noted that K = diag(σ12 , σ22 ) because of independence – and therefore uncorrelation – between X and Y. The matrix form of the pdf (3.54) is a bit more involved but only a small effort is required to show that we can write fX (x) =

1 exp − (x − m)T K−1 (x − m) 2 2π det(K) ,

1

(3.57)

, where det(K) = σ1 σ2 and K−1 = diag(1/σ12 , 1/σ22 ). From the vector X we can pass to the vector Z of standardized normal r.v.s by means of the linear transformation Z = S−1 (X − m), where S is the diagonal matrix introduced in eq. (3.33). By virtue of eq. (3.39b) the CF

The multivariate case

99

of Z is −1/2 vT R v

ϕZ (v) = e

*

+

1 = exp − (v12 + v22 ) 2

(3.58)

where R is the correlation matrix which, in our case of independent r.v.s, equals the identity matrix I = diag(1, 1). As expected, the CF (3.58) is the product of two standardized one-dimensional CFs and, clearly, the joint-pdf will also be in the form of product of two standardized one-dimensional pdfs, that is, fZ (z1 , z2 ) = fZ1 (z1 )fZ2 (z2 ) =

* + 1 2 1 exp − z1 + z22 2π 2

(3.59)

If, on the other hand, the two normal variables X, Y are correlated eqs (3.56) and (3.57) are still valid but it is understood that now K – and therefore K−1 – are no longer diagonal because K12 = Cov(X, Y) = 0. So, the explicit expression of the joint-CF becomes * + 1 2 2 ϕX (u, v) = exp i(u m1 + v m2 ) − σ1 u + σ22 v2 + 2K12 uv (3.60) 2 from which – by setting u = 0 or v = 0, as appropriate – it is evident that both marginal distributions are one-dimensional CFs of Gaussian random variables (see eq. (2.52)). Using eq. (3.57), the explicit form of the joint-pdf is written fX (x, y) =

1 e−γ (x,y)/2 , 2π σ1 σ2 (1 − ρ 2 )

(3.61a)

where ρ = K12 /σ1 σ2 is the correlation coefﬁcient between X and Y and the function γ (x, y) in the exponential is 1 γ (x, y) = 1 − ρ2

"

(x − m1 )2 σ12

# (x − m1 )(y − m2 ) (y − m2 )2 − 2ρ + σ1 σ2 σ22 (3.61b)

From eqs (3.60) and/or (3.61) we note an important property of jointly Gaussian random variables: the condition Cov(X, Y) = 0 – and therefore ρ = 0 if both σ1 , σ2 are ﬁnite and different from zero – is necessary and sufﬁcient for X and Y to be independent. In this case, in fact, the jointCF (pdf) becomes the product of two one-dimensional Gaussian CFs (pdfs). The equivalence of uncorrelation and independence for Gaussian r.v.s is noteworthy because – we recall from Section 3.3 – uncorrelation does not, in general, imply independence.

100 Probability theory We consider now another important result for jointly Gaussian r.v.s. Preliminarily, we notice that – referring to a three-dimensional system of coordinate axes x, y, z – the graph of the pdf,(3.61) is a bell-shaped surface with maximum of height z = (2π σ1 σ2 1 − ρ 2 )−1 above the point x = m1 , y = m2 . If we cut the surface with a horizontal plane (i.e. parallel to the x, y-plane), we obtain an ellipse whose projection on the x, y-plane has equation (x − m1 )2 σ12

− 2ρ

(x − m1 )(y − m2 ) (y − m2 )2 + = const. σ1 σ 2 σ22

(3.62)

which, in particular, is a circle whenever ρ = 0 and σ1 = σ2 . On the other hand, as ρ approaches +1 or −1, the ellipse becomes thinner and thinner and more and more needle-shaped until ρ = 1 or ρ = −1, when the ellipse degenerates into a straight line. In these limiting cases det(K) = 0, K−1 does not exist and one variable depends linearly on the other. In other words, we are no longer dealing with a two-dimensional random vector but with a single random variable and this is why one speaks of degenerate or singular Gaussian distribution. Returning to our main discussion, the important result is the following: when the principal axes of the ellipse are parallel to the coordinate axes, then ρ = 0 and the two r.v.s are uncorrelated – and therefore independent. In other words, by means of a rotation of the coordinate axes x, y it is always possible to pass from a pair of dependent Gaussian variables – whose pdf is in the form (3.61) – to a pair of independent Gaussian variables. This property can be extended to n dimensions and is frequently used in statistical applications (see the following chapters). Let us examine this property more closely. Given the ellipse (3.62), it is known from analytic geometry that the relation tan 2α =

2ρσ1 σ2 σ12 − σ22

(3.63)

determines the angles α between the x-axis and the principal axes of the ellipse (eq. (3.63) leads to two values α, namely α1 , α2 where α1 , α2 differ by π/2). If we rotate the x, y-plane through an angle α, the new coordinate axes are parallel to the ellipse principal axes and the cross-product term in (3.62) vanishes. If, in addition to this rotation, we perform now a rigid translation of the coordinate system which brings the origin to the point (m1 , m2 ), the ellipse will also be centered in the origin. At this point, the original pdf (3.61) has transformed into 1 p2 q2 f (p, q) = (3.64a) exp − 2 − 2π σp σq 2σq2 2σp

The multivariate case

101

where we call p, q the ﬁnal coordinate axes obtained by ﬁrst rotating and then rigidly translating the original axes x, y. Moreover, it can be shown that the new variances σp2 , σq2 are expressed in terms of the original variances by the relations σp2 = σ12 cos2 α + ρσ1 σ2 sin 2α + σ22 sin2 α σq2 = σ12 sin2 α − ρσ1 σ2 sin 2α + σ22 cos2 α

(3.64b)

, from which it follows σp σq = σ1 σ2 1 − ρ 2 . Equation (3.64b) is obtained by noticing that the (linear, since α is ﬁxed) relation between the x, y and the p, q axes – and therefore between the original random vector X = (X, Y)T and the new random vector P = (P, Q)T – is p cos α = q − sin α

sin α cos α

x m1 − m2 y

(3.65)

so that eq. (3.64b) follows by virtue of eq. (3.35). As a side comment, eqs (3.64b) represent the diagonal terms of the covariance matrix of the twodimensional vector P. By using eqs (3.35) and (3.63) the reader is invited to determine that, as expected, Cov(P, Q) = 0, that is, that the off-diagonal terms of the ‘new’ covariance matrix are zero. From (3.64a), if needed, it is then possible to take a further step and pass to the standardized Gaussian random vector Z whose pdf and CF are given by eqs (3.59) and (3.58), respectively. At this point we can consider a frequently encountered problem and determine the probability Pk that a point falls within the ellipse whose principal axes are k times the standard deviations σp , σq of the two variables. Calling Ek this ellipse (centered in the origin), the probability we are looking for is Pk = P[(P, Q) ∈ Ek ] =

f (p, q) dp dq

(3.66a)

Ek

where f (p, q) is given by eq. (3.64a). Passing to the standardized variables z1 = p/σp and z2 = q/σq the ellipse Ek becomes a circle Ck of radius k and 2 2 % $ z z 1 dz1 dz2 exp − 1 − 2 Pk = P Z12 + Z22 ≤ k2 = 2π 2 2

(3.66b)

Ck

The integral on the r.h.s. can now be calculated by turning to the polar coordinates z1 = r cos θ, z2 = r sin θ (recall from analysis that the Jacobian

102 Probability theory determinant of this transformation is |J| = r) and we ﬁnally get 1 Pk = 2π

2π k

−(r2 /2)

re 0 0

k dr dθ =

r e−(r

2

/2)

dr = 1 − exp(−k2 /2)

0

(3.66c) so that, for instance, we have the probabilities P1 = 0.393, P2 = 0.865 and P3 = 0.989 for k = 1, k = 2 and k = 3, respectively. This result is the twodimensional counterpart of the well-known fact that for a one-dimensional Gaussian variable X the probability of obtaining a value within k standard deviations is ⎧ ⎪ ⎨0.683 for k = 1 (3.67) P[|X − µX | ≤ kσX ] = 0.954 for k = 2 ⎪ ⎩ 0.997 for k = 3 Equation (3.67), in addition, can also be used to calculate the twodimensional probability to obtain a value of the vector (P, Q) within the rectangle Rk of sides 2kσp , 2kσq centered in the origin. In fact, since P and Q are independent, the two-dimensional probability P[(P, Q) ∈ Rk ] is given by the product of the one-dimensional probabilities (3.67); consequently P[(P, Q) ∈ R1 ] = (0.683)2 = 0.466, P[(P, Q) ∈ R2 ] = (0.954)2 = 0.910, etc. and it should be expected that P[(P, Q) ∈ Rk ] > P[(P, Q) ∈ Ek ] because the ellipse Ek is inscribed in the rectangle Rk . We close this rather lengthy section with a few general comments on Gaussian random vectors in any number of dimensions: (i) The property of being Gaussian is conserved under linear transformations (as the discussion above has shown more than once in the two-dimensional case). (ii) The marginal distributions of a jointly-Gaussian are individually Gaussian. However, the reverse may not be true and examples can be given of individually Gaussian r.v.s which, taken together, do not form a Gaussian vector. (iii) The CF and pdf of a jointly-Gaussian n-dimensional vector are written in matrix form as in eqs (3.56) and (3.57); in this latter equation, however, the factor 2π at the denominator becomes (2π )n/2 . √ In other words, at the denominator of (3.57) there must be a factor 2π for each dimension. (iv) Let us examine in the general case the possibility of passing from a Gaussian vector of correlated random variables (i.e. with a nondiagonal covariance matrix) to a Gaussian vector of independent – or even standardized – random variables (that is with a diagonal covariance

The multivariate case

103

matrix). In two-dimensional this was accomplished by ﬁrst rotating the coordinate axes and then translating the origin to the point (m1 , m2 ) but it is evident that we can ﬁrst translate the axes and then rotate them without changing the ﬁnal result. So, starting from a n-dimensional Gaussian vector X of correlated r.v.s X1 , . . . , Xn whose pdf is

1 1 T −1 fX (x) = (3.68a) exp − (x − m) K (x − m) , 2 (2π )n/2 det(K) -= we can translate the axes and consider the new ‘centered’ vector X X − m with pdf

1 1 T −1 ˆ = fX (3.68b) exp − xˆ K xˆ , - (x) 2 (2π )n/2 det(K) Now, since K is symmetric and positive deﬁnite (i.e. the Gaussian vector is assumed to be non-degenerate), a theorem of matrix algebra states that there exists a non-singular matrix H such that HHT = K. From this relation we get HHT K−1 = I and then HHT K−1 H = H which, in turn, implies HT K−1 H = I. Using this same matrix H let us now pass to the new random - = HZ. The term at the vector Z = (Z1 , . . . , Zn )T deﬁned by the relation X exponential of (3.68b) becomes xˆ T K−1 xˆ = (Hz)T K−1 Hz = zT HT K−1 Hz = zT Iz = zT z which is the sum of squares z12 + z22 + · · · + zn2 . Moreover, the Jacobian determinant of the transformation to Z is det(H) , so that the multiplying factor before the exponential becomes det(H)/ (2π )n det(K); however, from HT K−1 H = I we get,(det H)2 det(K−1 ) = 1 and since det(K−1 ) = [det(K)]−1 , then det(H) = det(K). Consequently, our ﬁnal result is

1 1 T z exp − z (3.69) fZ (z) = 2 (2π )n/2 which, as expected, is the pdf of a standardized Gaussian vector whose covariance matrix is I. Also, it is now clear that the matrix H represents a n-dimensional rotation of the coordinate axes.

3.4

More on conditioned random variables

The subject of conditioning has been discussed in both Chapters 1 and 2 (see, in particular, Section 2.5.2). Here we return on the subject for two main reasons: ﬁrst, because some more remarks are worthy of mention in their own right and, second, because a number of new aspects are due the developments of the preceding sections of this chapter.

104 Probability theory Let us start with some additional results to what has been said in Section 2.5.2 by working mostly in the real probability space (R, B, PX ). We do so because, as stated before, it is (R, B, PX ) which is considered in practice while the original space (W, S, P) is an entity in the background with only occasional interest in applications. Consider an absolutely continuous r.v. X deﬁned on (W, S, P). This, we recall, implies that the measure PX is absolutely continuous with respect to the Lebesgue measure on the real line and there exists a function fX (the pdf of X) such that PX (B) = P(X

−1

(B)) =

fX (x) dx B

Then, although the set C = {x} is indeed a Borel set of R, we cannot – at least in the usual way – deﬁne a conditional probability with respect to this event because PX (x) = 0. However, for h > 0 consider the Borel set Bh deﬁned as Bh = (x − h, x + h]. Then PX (Bh ) > 0 and we can condition on this event by deﬁning, for every A ∈ S, the measure PBh exactly as we did in Section 2.5.2 (eq. (2.60); again with a slight misuse of notation because Bh is not a set of S. Rigorously, we should write PX−1 (Bh ) ). Now, with a slight change of notation let us call PX (· | Bh ) its image measure in R instead of PX | Bh . Then, given B ∈ B we have PX (B | Bh ) =

PX (B ∩ Bh ) PX (Bh )

(3.70)

and at this point we can try to deﬁne PX (B | x) as the limit of PX (B | Bh ) as h → 0. By virtue of Bayes’ theorem (eq. (1.14)) together with the total probability formula of eq. (1.12) we can write PX (B | Bh ) as PX (B | Bh ) = PX (B) = PX (B)

PX (Bh | B) PX (Bh ) FX | B (x + h) − FX | B (x − h) FX (x + h) − FX (x − h)

(the second equality is due to the basic properties of the PDFs FX | B and FB ), then, dividing both the numerator and denominator by 2h and passing to the limit we get the desired result PX (B | x) = PX (B)

fX | B (x) fX (x)

(3.71)

provided that the conditional density fX | B exists. The fact that PX (B | x) is not deﬁned whenever fX (x) = 0 is not a serious limitation because the set

The multivariate case

105

N = {x : fX (x) = 0} has probability zero, meaning that, in practice, it is unimportant as far as probability statements are concerned. In fact PX (N) =

fX (x) dx = 0 N

If now in eq. (3.71) we move the factor fX (x) to the left-hand side and integrate both sides over the real line we get (since fX | B is normalized to unity) ∞ PX (B) =

fX (x) PX (B | x) dx

(3.72)

−∞

which, because of its analogy with eq. (1.12), is the continuous version of the total probability formula (see also eq. (3.80b)); the sum becomes now an integral and the probabilities of the conditioning events Aj are now the inﬁnitesimal probabilities fX (x) dx. In the case of discrete r.v.s the complications above do not exist. If AX is the (discrete) set of values taken on by the r.v. X and xi ∈ AX is such that PX {xi } = pX (xi ) > 0, then the counterpart of eq. (3.71) is PX (B | xi ) = PX (B)

pX | B (xi ) pX (xi )

(3.73)

and it is not deﬁned if p(xi ) = 0. On the other hand, the total probability formula reads PX (B) = pX (xi )PX (B | xi ) (3.74) xi ∈AX

We turn now to some new aspects of conditional probability brought about by the discussion of the previous sections. Consider a two-dimensional absolutely continuous random vector X = (X, Y) with joint-PDF FXY (x, y) and joint-pdf fXY (x, y); we want to determine the statistical description of, say, Y conditioned on a value taken on by the other variable, say X = x. We have now three image measures, PX , PY in R and the joint measure PXY (or PX ) in R2 ; all of them, however, originate from P in W. Therefore, if we look for a probabilistic description of an event relative to Y conditioned on an event relative to X it is reasonable to consider – in (W, S) – the conditional probability (where Jy = (−∞, y] and Bh is as above) P[Y −1 ( Jy ) | X −1 (Bh )] =

P[Y −1 ( Jy ) ∩ X −1 (Bh )] PXY ( Jy ∩ Bh ) = PX (Bh ) P(X −1 (Bh ))

106 Probability theory and deﬁne the conditional PDF FY | X (y | x) as the limit of this probability as h → 0. As before, we divide both the numerator and denominator by 2h and pass to the limit to get FY | X (y | x) =

1 fX (x)

∂FXY (x, y) ∂x

(3.75a)

Then, taking the derivative of both sides with respect to y we obtain the conditional pdf fY | X (y | x) =

1 fX (x)

fXY (x, y) ∂ 2 FXY (x, y) = fX (x) ∂y∂x

(3.75b)

A few remarks are in order: (a) the function fY | X is not deﬁned at the points x where fX (x) = 0; however, as noticed above, this is not a serious limitation; (b) fY | X is a function of y alone and not a function of the two variables x, y. In this case x plays the role of a parameter: for a given value, say x1 , we have a function fY | X (y | x1 ) and we have a different function fY | X (y | x2 ) for x2 = x1 ; (c) being a pdf in its own right, fY | X is normalized to unity. In fact, recalling eq. (3.10b) we get ∞ −∞

1 fY | X (y | x) dy = fX (x)

∞ fX (x, y) dy = 1 −∞

For the same reason it is clear that the usual relation between pdf and PDF holds, that is y FY | X (y | x) =

fY | X (t | x) dt −∞

(d) the symmetry between the two variables leads immediately to the conditional-pdf fX | Y (x | y) of X given Y = y, that is, fX | Y (x | y) =

fX (x, y) fY (y)

(3.76)

(e) if X is discrete and AX = {x1 , x2 , . . .}, AY = {y1 , y2 , . . .} are the ranges of X and Y, respectively, then the joint-pmf takes on values in AX × AY

The multivariate case

107

and the counterpart of (3.76) can be written as

pX | Y (xi | yk ) = P(X = xi | Y = yk ) =

pX (xi , yk ) pY (yk )

(3.77)

where it is assumed that pY (yk ) = 0.

In the light of the considerations above, we can now obtain some relations which are often useful in practical cases. We do so for the absolutely continuous case leaving the discrete case to the reader. First, by virtue of eqs (3.75b) and (3.76) we note that it is possible to express the joint-pdf of X in the two forms fXY (x, y) = fY | X (y | x)fX (x) fXY (x, y) = fX | Y (x | y)fY (y)

(3.78)

Then, combining these two results we get

fX | Y (x | y) = fY | X (y | x)

fX (x) fY (y)

(3.79)

and a similar equation for fY | X . Next, in order to obtain the counterpart of the total probability expression of eq. (3.72), we can go back to the probability P by letting B be the event Y −1 ( Jy ); then P(Y −1 ( Jy )) = FY (y) and we obtain the marginal-PDF of Y in terms of the conditional-PDF FY | X and of the pdf of the conditioning variable, that is, ∞ FY (y) =

FY | X (y | x)fX (x) dx

(3.80a)

−∞

Differentiating with respect to y on both sides leads to the total probability formula ∞ fY (y) =

fY | X (y | x)fX (x) dx −∞

(3.80b)

108 Probability theory (which could also be obtained from the second of (3.10b) using (3.79)). By symmetry, it is then evident that ∞ FX | Y (x | y)fY (y) dy

FX (x) = −∞

(3.81)

∞ fX (x) =

fX | Y (x | y)fY (y) dy −∞

If the expression (3.80b) for fY is inserted at the denominator of eq. (3.79) we note a formal analogy with Bayes’ theorem of eq. (1.14); for this reason eq. (3.79) – and its counterpart for fY | X – is also called Bayes’ formula (in the continuous case). In Section 3.2.1 we pointed out that two variables X, Y are independent if and only if fXY (x, y) = fX (x)fY (y). Therefore, by virtue of eq. (3.78), independence implies fY | X (y | x) = fY (y)

(3.82)

fX | Y (x | y) = fX (x)

as it might be expected considering that knowledge of a speciﬁc outcome, say X = x, gives no information on Y. At this point, the extension to more than two r.v.s is immediate and we a multionly mention it brieﬂy here, leaving the rest to the reader. If we call X dimensional vector of components X1 , . . . , Xm , Y1 , . . . , Yn with joint-pdf fX then, for example, fX | Y (x1 , . . . , xm | y1 , . . . , yn ) =

fX (x1 , . . . , xm , y1 , . . . , yn ) fY (y1 , . . . , yn )

(3.83)

where we denoted by fY the marginal-pdf relative to the n Y-type variables. Similarly, the generalization of eq. (3.80b) becomes fY (y) =

fY | X (y | x)fX (x) dx

(3.84)

Rm

where fX the marginal-pdf of the X variables and dx = dx1 · · · dxm . As an exercise to close this section, we also invite the reader to examine the case of a two-dimensional vector X = (X, Y) where X is absolutely continuous with pdf fX (x) and Y is discrete with pmf deﬁned by the values pY (yi ).

The multivariate case 3.4.1

109

Conditional expectation

As noted in Section 2.5.2, the theory deﬁnes conditional expectations as abstract Lebesgue integrals in W with respect to an appropriate conditional measure which, in turn, depends on the conditioning event and is ultimately expressed in terms of the original measure P. In practice, however, owing to the relation between measures in W and their image measures (through a random variable or a random vector), expectations become in the end Lebesgue–Stieltjes integrals on R, R2 or Rn , whichever is the case. These integrals, in turn, are sums or ordinary Lebesgue integrals (i.e. Riemann integrals in most applications) depending on the type of distribution function. Owing to the developments of the preceding section, it should be expected that the conditional expectation of X given the event Y = y is expressed as E(X | y) =

x dFX | Y

(3.85)

R

which, in the absolutely continuous case becomes ∞ E(X | y) =

xfX | Y (x | y) dx = −∞

1 fY (y)

∞ xfX (x, y) dx

(3.86)

−∞

where in the second equality we took eq. (3.76) into account. It is understood that analogous relations hold for E(Y | x). On the other hand, in the discrete case (conditioning on the event Y = yk ) we have E(X | yk ) =

xi pX | Y (xi | yk ) =

1 xi pX (xi , yk ) pY (yk )

(3.87)

i

all i

More generally, if g is a measurable function of both X and Y we have the fundamental relations (their discrete counterparts are left to the reader) ∞ E[g(X, Y) | y] =

g(x, y)fX | Y (x | y) dx −∞

(3.88)

∞ E[g(X, Y) | x] =

g(x, y)fY | X (y | x) dy −∞

It is evident that eq. (3.86) coincides with the ﬁrst of (3.88) when g(x, y) = x and also that the expressions for all conditional moments can be obtained as special cases of eq. (3.88), depending on which one of the two variables is the conditioning one.

110 Probability theory Being based on the properties of the integral, conditional expectations satisfy all the properties of expectation given in Chapter 2. In particular we mention, for example (i) the conditional expectation of a constant is the constant itself; (ii) if a, b are two constants and X, Y1 , Y2 are random variables then linearity holds, that is, E(aY1 + bY2 | x) = aE(Y1 | x) + bE(Y2 | x); (iii) if Y1 ≤ Y2 then E(Y1 | x) ≤ E(Y2 | x). Now, so far we have spoken of conditional expectations of, say, X given Y = y by tacitly assuming that y is a given, well-speciﬁed value. If we adopt a more general point of view we can look at expectations as functions of the values taken on by the random variable Y. In other words since, in general, we have a value of E(X | y) for every given y we may introduce the real-valued function g(Y) ≡ E(X | Y) deﬁned on the range of Y. This function – which, clearly, takes on the value E(X | y) when Y = y – can be shown to be measurable and therefore it is a random variable itself. In this light it is legitimate to ask about its expectation E[g(Y)] = E[E(X | Y)]. The interesting result is that we get E[E(X | Y)] = E(X)

(3.89a)

and, by symmetric arguments E[E(Y | X)] = E(Y)

(3.89b)

In fact, in the absolutely continuous case, for example,

E[E(X | Y)] =

E(X | Y)fY (y) dy =

=

xfX (x, y) dx dy =

xfX | Y (x | y) dx fY (y) dy

x

fX (x, y) dy

dx

=

xfX (x) dx = E(X)

(all integrals are from −∞ to +∞ and eqs (3.76) and (3.10b) have been taken into account). Equations (3.89) – which are sometimes useful in practice – may appear confusing at ﬁrst sight but they state a reasonable fact: for instance, eq. (3.89a) shows that E(X) can be calculated by taking a weighted average on all the expected values of X given Y = y, each term being weighted by the probability of that particular conditioning event Y = y.

The multivariate case

111

Equation (3.89a) can be generalized to E[g(X)] = E[E(g(X) | Y)]

(3.90)

where g(X) is a (measurable) function of X. With the appropriate modiﬁcations, the same obviously applies to (3.89b). By similar arguments, the reader is invited to prove that Var(X) = E[Var(X | Y)] + Var[E(X | Y)]

(3.91)

Var(Y) = E[Var(Y | X)] + Var[E(Y | X)]

(Hint: to prove the ﬁrst of (3.91) start from Var(X | Y) = E(X 2 | Y) − E2 (X | Y) and use eq. (3.90). For our purposes, the discussion above sufﬁces. However, for the interested reader we close this section with some additional remarks of theoretical nature on the function E(X | Y). We simply outline the general ideas and more details can be found in the references at the end of the chapter. Consider an event G ∈ S. As a consequence of eq. (2.58), the expectation of a r.v. X conditioned on G can be written as 1 1 E(X | G) = X dPG = IG X dP = XdP (3.92a) P(G) P(G) W

W

G

which leads to P(G)E(X | G) =

X dP

(3.92b)

G

This last expression makes no reference to the conditional measure PG and can be assumed to be the deﬁning relation of E(X | G). Clearly, in the same = {∅, G, GC , W} is a way one can deﬁne E(X | GC ). Then, noting that G ⊂ S (the σ -algebra generated by G) one can deﬁne a function σ -algebra G on G as E(X | G) = E(X | G)IG + E(X | GC )IGC E(X | G)

(3.93)

is a simple function (see the deﬁnition of simple function in E(X | G) Appendix B) which is measurable – and therefore a random variable – with and it is such that, for every set A ∈ G respect to both S and G A

dP = E(X | G)

X dP A

(3.94)

112 Probability theory (in this case A can only be one of the four sets ∅, G, GC , W; using the deﬁnition of integral for simple functions, the reader is invited to verify eq. (3.94)). If, in particular X = IF (where F ∈ S) then the r.h.s. of (3.94) equals P(F ∩A). then by virtue of the Radon– Setting PF (A) = P(F ∩ A) for every A ∈ G Nikodym theorem we can deﬁne the conditional probability as a special case of conditional expectation, that is, = E(IF | G) P(F | G)

(3.95)

which agrees with the fact that the measure of a set is the expectation of its indicator function. Now, besides this illustrative example, it can be shown that this same line of reasoning extends to any σ -algebra S ⊂ S and the resulting function E(X | S) is called the conditional expectation of X given S. In particular, if S is the σ -algebra generated by a collection of sets G1 , G2 , . . . , Gn ∈ S such that W = ∪ni=1 Gi , then E(X | S) is a r.v. on (W, S, P) which takes on the S. Also, one can value E(X | Gi ) on Gi and satisﬁes eq. (3.94) for every A ∈ deﬁne P(F | S) as above. However, it is not necessary for S to be determined by a ﬁnite collection of sets. Therefore, if Y is another r.v. deﬁned on the space (W, S, P), for every Borel set B ⊂ R one can consider the σ -algebra Y generated by the inverse images Y −1 (B) and introduce the function E(X | Y) which satisﬁes the counterpart of (3.94), that is,

dP = E(X | Y)

Y −1 (B)

X dP

(3.96)

Y −1 (B)

is constant on every set of the form Since it can be shown that E(X | Y) is a Y −1 (y) (where y is a ﬁxed value in R), then it follows that E(X | Y) function of Y which takes on the value E(X | y) for all the elements w ∈ W = E(X) such that w ∈ Y −1 (y). Equation (3.96) then shows that E[E(X | Y)] which, on more theoretical grounds, justiﬁes eq. (3.89a). 3.4.2

Some examples and further remarks

In order to illustrate with an example the considerations of the preceding two sections we start with the bivariate Gaussian distribution. If the two variables X,Y are correlated their joint-pdf is given by eqs (3.61a) and (3.61b). We could obtain the marginal-pdfs by using eq. (3.10b) but it is quicker to consider the joint-CF of eq. (3.60) and note that the marginal CFs are both one-dimensional Gaussian. It follows that fX (x) and fY (y) are Gaussian pdfs with parameters E(X) = m1 , Var(X) = σ12 and E(Y) = m2 , Var(Y) = σ22 , respectively. For the conditional pdfs we can use eq. (3.78) so that, say, fY | X (y | x) is given by fY | X (y | x) = fXY (x, y)/fX (x). Explicitly, after some

The multivariate case

113

manipulations we get fY | X (y | x) =

,

1

σ2 2π(1 − ρ 2 )

exp[−h(x, y)]

(3.97a)

where the function in the exponential is 1 h(x, y) = 2(1 − ρ 2 )

x − m1 y − m2 −ρ σ2 σ1

2 (3.97b)

and can be rewritten in the form h(x, y) =

1 2σ22 (1 − ρ 2 )

2

y − m2 − ρ

σ2 (x − m1 ) σ1

(3.97c)

from which it is evident that the conditional expectation and variance are mY | X = E(Y | X) = m2 + ρ

σ2 (x − m1 ) σ1

(3.98)

σY2 | X = Var(Y | X) = σ22 (1 − ρ 2 ) Equation (3.98) show that (i) as a function of x, the conditional expectation of Y given x is a straight line (which is called the regression line of Y on X) and (ii) the conditional variance does not depend on x. With the obvious modiﬁcations, relations similar to (3.97) and (3.98) hold for fX | Y (x | y), E(X | Y) and Var(X | Y). If the two variables are independent – which, we recall, is equivalent to uncorrelated for the Gaussian case – then the conditional-pdfs coincide with the marginal pdfs and the conditional parameters coincide with the unconditioned ones. Equations (3.97) and their counterparts for fX | Y (x | y), in addition, show that the conditional-pdfs of jointly Gaussian r.v.s are Gaussian themselves. We do not prove it here but it can be shown that this is an important property which extends to the n-dimensional case: all the conditional pdfs that can be obtained from a jointly Gaussian vector are Gaussian. If now, as another example, we consider the joint-pdf of Example 3.1(b) (eq. (3.13a)), the reader is invited to determine that

1 1 fX | Y (x | y) = √ exp − (x + y/2)2 3 3π

(3.99)

E(X | Y) = −y/2 and also that E(XY) = −1. So, if we note from the marginal-pdfs (3.14a) and (3.14b) that E(X) = E(Y) = 0 and Var(X) = Var(Y) = 2, it follows from eqs (3.19a) and (3.22) that Cov(X, Y) = −1 and ρXY = −1/2.

114 Probability theory In the bivariate Gaussian case above we spoke of regression line of Y on X. In the general case of a non-Gaussian pdf E(Y | X) – as a function of x – may not be a straight line and then one speaks of regression curve of Y on X. For some non-Gaussian pdfs, however, it may turn out that E(Y | X) is a straight line, that is, that we have E(Y | X) =

y fY | X (y | x) dy = a + bx

(3.100)

where a and b are two constants. Now, since E[E(Y | X)] = E(Y) we can take eq. (3.100) into account to get E(Y) =

y fXY (x, y) dx dy =

fX (x)

y fY | X (y | x) dy

dx

fX (x)(a + bx) dx = a + bE(X)

=

(3.101)

showing that the regression line passes through the point (E(X), E(Y)). By similar arguments, we also obtain (the easy calculations are left to the reader) E(XY) = aE(X) + bE(X 2 )

(3.102)

so that, in the end, the slope and intercept of the straight line are given by b=

E(XY) − E(X)E(X) Cov(X, Y) = 2 2 Var(X) E(X ) − E (X)

(3.103a)

a = E(Y) − bE(X) where eqs (3.19a) and (3.19b) have been taken into account in the second equality for b. By substituting eqs (3.103a) in (3.100) and recalling eq. (3.22) we see that in all cases where E(Y | X) is a linear function of x the ﬁrst of eq. (3.98) holds. On the other hand, now the second of (3.98) may no longer hold and σY2 | X

=

{y − E(Y | X)}2 f (y | x) dy

is, in general, a function of x. Nonetheless, the quantity σ22 (1 − ρ 2 ) still has a meaning: it represents a measure of the average variability of Y around the regression line on X. In fact, it is left to the reader to show that by deﬁning 2 σY(avg) as the weighted (with the probability density of the x-values) average

The multivariate case of

σY2 | X ,

115

then

2 σY(avg)

≡

σY2 | X fX (x) dx = σ22 (1 − ρ 2 )

(Hint: use (3.100) and the second of (3.103a) to determine that [y − E(Y | X)]2 = (y − E(Y))2 + b2 (x − E(X))2 − 2b(y − E(Y))(x − E(X)), insert in the expression of σY2 | X and then use eq. (3.75b) and the ﬁrst of (3.103a).) 2 From the expression of σY(avg) we note, however, that the second of eq. (3.98) holds whenever σY2 | X does not depend on x. In all these particular cases we 2 have σY(avg) = σY2 | X = σ22 (1 − ρ 2 ). The bivariate Gaussian – as we have seen – is one of these cases. If now, in addition to (3.100), we also assume that E(X | Y) = c + dy then d=

Cov(X, Y) Var(Y)

(3.103b)

a = E(X) − dE(Y) and the geometric mean of b and d is the correlation coefﬁcient, that is, √

bd =

Cov(X, Y) = ρXY σX σ Y

(3.104)

which, as noted in Section 3.3, is a measure of the extent of the linear relationship between the two variables. As a ﬁnal remark we point out that the fact that E(Y | X) is a linear function of x does not necessarily imply, in general, that E(X | Y) is a linear function of y – and conversely; the bivariate Gaussian distribution is, in this respect, an exception. More on linear regression in statistical applications is delayed to Chapter 7.

3.5

Functions of random vectors

It often happens that we have some information on one or more random variables but our interest – rather than in the variables themselves – lies in a function of these variables. In Section 2.5.3, we already touched this subject by considering mainly the one-dimensional case; we now move on from there extending the discussion to random vectors. Let X = (X1 , . . . , Xn ) be a n-dimensional random vector and let Z = (Z1 , . . . , Zn ) be such that Z = g(X), where this symbol means that the

116 Probability theory function g : Rn → Rn has components g1 , . . . , gn and Z1 = g1 (X1 , . . . , Xn ) Z2 = g2 (X1 , . . . , Xn ) .. .

(3.105)

Zn = gn (X1 , . . . , Xn ) First of all we note that Z is a random vector if all the gk (k = 1, . . . , n) are Borel functions, a condition which is generally true in most practical cases. If we suppose further that we are dealing with absolutely continuous vectors and that the joint-pdf fX (x) is known, we can obtain fZ (z) by using a well-known change-of-variables theorem of analysis. This result leads to an equation formally similar to (2.71), that is, fZ (z) = fX (g−1 (z))| det(J)|

(3.106a)

where J is the Jacobian matrix ⎡ −1 ∂g1 /∂z1 ⎢∂g −1 /∂z1 ⎢ 2 J=⎢ .. ⎣ . ∂gn−1 /∂z1

∂g1−1 /∂z2 ∂g2−1 /∂z2 .. . ∂gn−1 /∂z2

⎤ . . . ∂g1−1 /∂zn . . . ∂g2−1 /∂zn ⎥ ⎥ ⎥ .. ... ⎦ . −1 . . . ∂gn /∂zn

(3.106b)

The assumptions of the theorem require that (a) g is one-to-one (so that X1 = g1−1 (Z1 , . . . , Zn ); X2 = g2−1 (Z1 , . . . , Zn ), etc.); (b) all the derivatives are continuous; (c) det( J) = 0. If one (or more) of the gk (k = 1, . . . , n) is not invertible (i.e. not one-toone), one needs to divide the domains of X and Z in a sufﬁcient number – say p – of mutually disjoint subdomains in such a way that – in these subdomains – there exists a one-to-one mapping between the two variables. Then eq. (3.106a) holds in each subdomain and the ﬁnal result fZ (z) is obtained by summing the p contributions. All the results above can also be used if Z is m-dimensional, with m < n. In this case one introduces n–m auxiliary variables and proceeds as stated by the theorem. Provided that the requirements of the theorem are satisﬁed, the choice of the auxiliary variables is arbitrary; therefore it is understood that one should choose them in a way that keeps the calculations as simple as possible. So, for instance, if X = (X1 , X2 ) is a two-dimensional vector and Z = g(X1 , X2 ) is one-dimensional, we can introduce the auxiliary variable

The multivariate case

117

Z2 = X2 ; then, the transformation (3.105) is Z1 = g(X1 , X2 )

(3.107a)

Z2 = X2 and * det( J) = det

∂g −1 /∂z1 0

+ ∂g −1 ∂g −1 /∂z2 = 1 ∂z1

(3.107b)

Consequently, eq. (3.106) reads fZ (z1 , z2 ) = fX (g

−1

∂g −1 (z1 ), x2 ) ∂z1

(3.107c)

and the desired result – that is, the marginal pdf of Z1 – is given by ∞ fZ1 (z1 ) =

fZ (z1 , z2 ) dz2

(3.107d)

−∞

Example 3.3 Let Z = X1 + X2 . As above, we introduce the auxiliary variable Z2 = X2 . Then X1 = Z1 − Z2 X2 = Z2 and det( J) = 1. Therefore fZ (z1 , z2 ) = fX (z1 − z2 , z2 ) and ∞ fZ1 (z1 ) =

∞ fX (z1 − z2 , z2 ) dz2 =

−∞

fX (z1 − x2 , x2 ) dx2

(3.108)

−∞

If the two variables X1 , X2 are independent with pdfs f1 (x1 ), f2 (x2 ), respectively, then fX (x1 , x2 ) = f1 (x1 )f2 (x2 ) and (3.108) becomes ∞ fZ1 (z1 ) =

f1 (z1 − x2 )f2 (x2 ) dx2

(3.109)

−∞

which is called the convolution integral of f1 and f2 ; this is a frequently encountered type of integral in applications of Physics and Engineering and is often denoted by the symbol f1 ∗ f2 . (Incidentally, we note that eq. (3.109) is in agreement with eq. (3.29) on CFs; in fact the Fourier transform of a

118 Probability theory convolution integral is given by the product of the individual Fourier transforms of the functions appearing in the convolution.) So, for instance, if X1 , X2 are independent and are both uniformly distributed in such a way that ' 1/(b − a), a < x ≤ b f1 (x) = f2 (x) = 0, otherwise b eq. (3.109) gives fZ1 (z1 ) = (b − a)−1 a f1 (z1 − x2 ) dx2 where z1 ranges from a minimum value of 2a to a maximum of 2b and is zero otherwise. The integral can be divided into two parts considering that (i) if z1 − x2 > a then x2 < z1 − a and (ii) if z1 − x2 < b then x2 > z1 − b. In the ﬁrst case we have 1 fZ1 (z1 ) = (b − a)2

z1 −a

dx2 = a

z1 − 2a (b − a)2

(3.110a)

which holds for 2a < z1 ≤ a + b (the second inequality is due to the fact that we must have z1 − a ≤ b, therefore z1 ≤ a + b). In the second case 1 fZ1 (z1 ) = (b − a)2

b dx2 = z1 −b

2b − z1 (b − a)2

(3.110b)

which holds for a + b < z1 ≤ 2b (z1 − b > a implies z1 > a + b). The distribution given by eqs (3.110a) and (3.110b) is called Simpson’s distribution. If, turning to another case, X1 , X2 are jointly-Gaussian and not independent (eqs (3.61a) and (3.61b)) then it can be shown (Refs [3, 4, 6, 17]) that the pdf of the r.v. Z = X1 + X2 is 1 (z − m1 − m2 )2 fZ (z) = ) exp − 2(σ12 + 2ρσ1 σ2 + σ22 ) 2π(σ12 + 2ρσ1 σ2 + σ22 ) (3.111) which is also Gaussian. The reverse statement, in general, is not true and the fact that Z = X1 + X2 is Gaussian does not necessarily imply that X1 , X2 are individually Gaussian. It does, however, if X1 , X2 are independent (Cramer’s theorem). In this case, fZ (z) is obtained by simply setting ρ = 0 in eq. (3.111). All these considerations on jointly-Gaussian vectors extend to n dimensions and the sum Z = ni=1 Xi of n Gaussian r.v.s is itself Gaussian with mZ = 2 mi and Var(Z) = σZ2 = σ if the Xi are independent and mZ = i 2 i i 2 = m and σ σ + 2 ρ i i i i i<j ij σi σj if they are not independent (ρij is the Z correlation coefﬁcient between Xi and Xj ).

The multivariate case

119

Example 3.4(a) Consider the random variable Z1 = X1 X2 . If we deﬁne Z2 = X2 then X1 = Z1 /X2 and | det( J)| = 1/|x2 |. Therefore ∞ fZ1 (z1 ) = −∞

1 fX (z1 /x2 , x2 ) dx2 |x2 |

(3.112)

Example 3.4(b) If, on the other hand, we consider the ratio Z1 = X1 /X2 – and, as above, we deﬁne Z2 = X2 – then X1 = Z1 X2 and | det(J)| = |x2 |. Therefore ∞ fZ1 (z1 ) =

|x2 |fX (z1 x2 , x2 ) dx2

(3.113a)

−∞

In addition, if the two original r.v.s are independent with pdfs f1 (x1 ), f2 (x2 ) ∞ fZ1 (z1 ) =

0 x2 f1 (z1 x2 )f2 (x2 ) dx2 −

x2 f1 (z1 x2 )f2 (x2 ) dx2 (3.113b)

−∞

0

So, for instance, if X1 , X2 are independent Gaussian r.v. with m1 = m2 = 0 and Var(X1 ) = σ12 , Var(X2 ) = σ22 then the term at the exponentials in both integrals of eq. (3.113b) can be written as x22 z12 σ22 + σ12 a = −x22 − 2 2 2 b σ 1 σ2 where we deﬁned a = z12 σ22 + σ12 and b = 2σ12 σ22 . Eq. (3.113b) then becomes ⎧∞ ⎫ 0 ⎨ ⎬ 1 x2 exp(−ax22 /b) dx2 − x2 exp(−ax22 /b) dx2 fZ1 (z1 ) = ⎭ 2π σ1 σ2 ⎩ −∞

0

and performing the change of variable t = ax22 /b so that (b/2a) dt = x2 dx2 we get 2 fZ1 (z1 ) = 2π σ1 σ2

b 2a

∞ 0

e−t dt =

σ 1 σ2 2 π(z1 σ22 + σ12 )

(3.114)

where we took into account that the two integrals within braces are equal to twice the integral in dt from 0 to ∞ and we substituted the explicit expressions for a and b to obtain the ﬁnal term on the r.h.s. of (3.114). The

120 Probability theory pdf of eq. (3.114) is a form of the so-called Cauchy distribution. In particular, if X1 , X2 are (independent) standardized r.v.s, then σ1 = σ2 = 1 and fZ1 (z1 ) =

1 π(z12

(3.115)

+ 1)

which is the form of the Cauchy distribution commonly found in the literature.

3.5.1

Numerical descriptors of functions of random variables

In the preceding section we determined how to obtain the probability distribution of a random variable (vector) which is a function of another random variable (vector) when we know the distribution of the original r.v. Depending on the functional relation between the two variables (vectors), this may not always be an easy task. It often happens, however, that the analyst’s interest lies in the numerical descriptors of Z = g(X) rather than in a complete probabilistic description of Z (i.e. fZ or FZ ). Moreover, in most cases one is mainly interested in the ﬁrst and second order moments of Z. These quantities can be obtained – or, more generally, approximated – without going through the determination of fZ or FZ . Starting from the case in which Z is one-dimensional we have already considered (Propositions 2.13 and 2.15; see also eq. (2.35c)) the situation when Z is a linear function of X, that is, Z = aX + b where a, b are two constants. Then E(Z) = aE(X) + b and Var(Z) = a2 Var(X), which, in turn, are special cases of the more general relations

E(Z) = Var(Z) = =

n i=1 n i=1 n i=1

ai E(Xi ) + b a2i Var(Xi ) + 2

ai aj Cov(Xi , Xj )

(3.116)

i<j

a2i Var(Xi ) +

ai aj Cov(Xi , Xj )

ij(i =j)

which occur whenever Z is a linear function of more than one r.v., that is, when Z = ni=1 ai Xi + b. If, in addition, the variables X1 , . . . , Xn are pairwise uncorrelated (or, more strictly, independent), the second of (3.116) becomes Var(Z) = i a2i Var(Xi ). Before turning to the general discussion, consider for instance the frequently encountered non-linear case Z = XY. Then, by the properties of

The multivariate case

121

covariance (Proposition 2.15 or eq. (3.19a)) we have mZ ≡ E(Z) = E(XY) = E(X)E(Y) + Cov(X, Y)

(3.117)

which becomes E(Z) = E(XY) = E(X)E(Y) whenever X, Y are uncorrelated or independent. Moreover if X, Y are independent it is left to the reader to show that the variance of Z is given by Var(Z) = Var(X)Var(Y) + E2 (X)Var(Y) + E2 (Y)Var(X) = σX2 σY2 + m2X σY2 + m2Y σX2

(3.118)

(Hint: start from the deﬁnition Var(Z) = E[(Z − mZ )2 ] and then take into account that X 2 , Y 2 are also independent r.v.s.) Let us now tackle the general problem. We will do so in three steps: in the order (a) a one-dimensional variable function of another one-dimensional variable, (b) a one-dimensional variable function of a random vector and (c) a random vector function of another random vector. Let now Z = g(X) where both X and Z are assumed to be absolutely continuous. If the function g is invertible then we have E(Z) =

z fZ (z) dz =

g(x)fX (x) dx

(3.119)

because z = g(x), dz = g (x) dx (the prime indicates the derivative) and, from eq. (2.71), fZ (z) = fX (x)/g (x) since dg −1 (z)/dz = 1/g (x). However, we can expand g(x) in a Taylor series around mX as z = g(x) = g(mX ) + (x − mX )g (mX ) +

1 (x − mX )2 g (mX ) + · · · 2 (3.120)

and insert this expression in (3.119) to get the approximate relation E(Z) ∼ = g(mX ) +

1 g (mX ) Var(X) 2

(3.121)

because it is easily veriﬁed that the term with the ﬁrst derivative yields zero in the integration. The calculation of the variance is a bit more involved. Similarly to eq. (3.119) we can write Var(Z) =

2

(z − mZ ) fZ (z) dz =

[g(x) − mZ ]2 fX (x) dx

(3.122)

122 Probability theory and use (i) the Taylor expansion (3.120) to approximate g(x) and (ii) eq. (3.121) to approximate mZ = E(Z). After a few passages we arrive at Var(Z) ∼ = [ g (mX )]2 Var(X) +

1 [ g (mX )]2 {M4 − Var2 (X)} 4

+ g (mX )g (mX )M3

(3.123a)

where we denoted by M3 and M4 the third and fourth-order central moments of X, respectively (i.e. M3 = E[(X − mX )3 ] and M4 = E[(X − mX )4 ]; also, using this notation note that Var(X) = M2 ). If the pdf fX (x) is symmetric about the mean, then M3 = 0 and if, in addition, it is Gaussian then (eq. (2.42d)) M4 = 3 M22 = 3 Var2 (X); therefore Var(Z) ∼ = [g (mX )]2 Var(X) +

1 [g (mX )]2 Var2 (X) 2

(3.123b)

For approximation purposes, one may sometimes use mZ = g(mX ) – which is equivalent to interchanging the expectation operator with the functional dependence, that is, E[g(X)] = g[E(X)] – for the mean and σZ2 = [g (mX )]2 σX2 for the standard deviation; however, it should be kept in mind that these relations are exact only in case of a linear relation between X and Z. Let now Z be a function of n random variables X1 , . . . , Xn , that is, Z = g(X1 , . . . , Xn ). In this case the linear approximation is frequently used; in other words one assumes that (i) the mean of the function equals the function of the X-means m1 , . . . , mm and (ii) the variance of the function depends only on the ﬁrst derivatives of g and on the variances σ12 , . . . , σn2 of X1 , . . . , Xn . Although this may seem a rather crude approximation, it generally leads to acceptable result and consequently – besides speciﬁc applications where a higher accuracy is required – linearization is the main technique to deal with the case Z = g(X1 , . . . , Xn ). So, linearizing the function g in a neighbourhood of m1 , . . . , mm we have g(x) = g(m) +

n ∂g (xi − mi ) + · · · ∂xi x=m

(3.124)

i=1

so that inserting this expression in E(Z) =

g(x) fX (x) dx

(3.125)

all the terms with the ﬁrst derivatives go to zero in the integration and mZ = E(Z) ∼ = g(m) = g(m1 , m2 , . . . , mn )

(3.126)

The multivariate case

123

Equation (3.126), in turn, can be used together with (3.124) in the expression Var(Z) =

[g(x) − mZ ]2 fX (x) dx

(3.127)

to arrive at the (approximate) result σZ2 = Var(Z) ∼ =

n

[Di g(m)]2 σi2 +

[Di g(m)][Dj g(m)] Kij (3.128a)

i,j; i =j

i=1

where, for short, we denoted Di g(m) = ∂g/∂xi |x=m . If the variables X1 , . . . , Xn are uncorrelated then Var(Z) ∼ =

n

[Di g(m)]2 σi2

(3.128b)

i=1

which is, nonetheless, an approximation due to the fact that we retained only the ﬁrst-order terms in the Taylor expansion. Introducing the column matrix D whose elements are the ﬁrst-order derivatives of g calculated at x = m, that is, ⎡

⎤ D1 g(m) ⎢D2 g(m)⎥ ⎢ ⎥ D=⎢ ⎥ .. ⎣ ⎦ . Dn g(m) then eq. (3.128a) can be concisely written in matrix form as Var(Z) ∼ = DT KD

(3.128c)

where K is the covariance matrix introduced in eq. (3.32a). If, in addition, the variables X1 , . . . , Xn are uncorrelated then K = diag(σ12 , . . . , σn2 ) and eq. (3.128c) reduces to the sum of squares of eq. (3.128b). A better approximation to E(Z) and Var(Z) than eqs (3.126) and (3.128), respectively, can be obtained by retaining the next term in the Taylor expansion (3.124). This term contains the second-order derivatives of g and can be written as n 1 2 1 2 Di g(m)(xi − mi )2 + Dij g(m)(xi − mi )(xj − mj ) 2 2 i=1

i,j; i =j

124 Probability theory where D2i g(m) = ∂ 2 g/∂x2i |x=m and D2ij g(m) = ∂ 2 g/∂xi ∂xj |x=m . In this approximation we are led to 1 2 1 2 Dij g(m) Kij Di g(m) σi2 + mZ ∼ = g(m) + 2 2

(3.129a)

i,j; i =j

i

or, if the variables are uncorrelated 1 2 Di g(m) σi2 mZ ∼ = g(m) + 2

(3.129b)

i

For the variance we can limit the calculations to the independent (or uncorrelated) case – although eq. (3.128b) will, in general, sufﬁce in this circumstance – and arrive at the rather lengthy relation σZ2 ∼ =

%2 1 $ 2 [Di g(m)]2 σi2 + Di g(m) {M4 (Xi ) − Var2 (Xi )} 4 i i $ % % $ + D2ij g(m) σi2 σj2 [Di g(m)] D2i g(m) M3 (Xi ) + i=j

i

(3.130) where we denoted by M3 (Xi ), M4 (Xi ) the third and fourth-order central moments of the variable Xi , respectively. As an example, we can return to the case Z = XY (X and Y independent) considered above. The reader can check that the approximation (3.128b) does not lead to the correct result (3.118) while, on the other hand, eq. (3.130) does. Finally, we examine now the most general case of m r.v.s Z1 , . . . , Zm which are functions of n r.v.s X1 , . . . , Xn . The situation is as follows Z1 = g1 (X1 , . . . , Xn ) Z2 = g2 (X1 , . . . , Xn ) .. .

(3.131)

Zm = gm (X1 , . . . , Xn ) m the means of Z1 , . . . , Zm , the linear approximation Denoting by m 1, . . . , m immediately yields m k ∼ = gk (m1 , . . . , mn ),

k = 1, 2, . . . , m

(3.132)

of the Z-variables is given by while the covariance matrix K ∼ K = DT KD

(3.133a)

The multivariate case

125

where K is the covariance matrix of the X-variables and we denoted by D the n × m matrix of derivatives ⎤ ⎤ ⎡ ⎡ ∂g1 /∂x1 ∂g2 /∂x1 . . . ∂gm /∂x1 D11 D21 . . . Dm1 ⎢∂g1 /∂x2 ∂g2 /∂x2 . . . ∂gm /∂x2 ⎥ ⎢D12 D22 . . . Dm2 ⎥ ⎥ ⎥ ⎢ ⎢ D=⎢ ⎥ = ⎢ .. .. .. ⎥ .. .. .. .. .. ⎦ ⎣ ⎣ . . ⎦ . . . . . . ∂g1 /∂xn ∂g2 /∂xn . . . ∂gm /∂xn D1n D2n . . . Dmn and it is understood that all derivatives are calculated at the point x = m. So, the (i, j)th element of the matrix is ij = Cov(Zi , Zj ) ∼ K =

Di k Kk l Dj l

k, l

(3.133b)

ji . being a covariance matrix, is clearly symmetric, that is, K ij = K and K, Clearly, eq. (3.133b) could also be directly obtained from the deﬁnition of covariance. In fact, for example, if Z1 = g1 (X1 , X2 ) and Z2 = g2 (X1 , X2 ) we have 12 = Cov(Z1 , Z2 ) = (z1 − m 1 )(z2 − m 2 ) fZ (z) dz K and by a similar line of reasoning as above we can expand both g1 , g2 in a neighborhood of m and use this expansion together with eq. (3.132) to get 12 ∼ K =

∂g1 ∂g2 Kkl ∂xk ∂xl k,l

which, as expected, is the same as eq. (3.133b). As the next example will show, a ﬁnal point worthy of notice is that independence of the X-variables does not, in general, imply independence of the Z-variables. Example 3.5 Let X1 , X2 be two uncorrelated r.v.s with variances K11 = σ12 , K22 = σ22 . Also let Z1 = 2X1 + X2 and Z2 = 5X1 + 3X2 . Then *

2 D KD = 5 T

+"

1 3

σ12

0

0

σ22

#*

2 1

+ " 4 σ12 + σ22 5 = 3 10 σ12 + 3σ22

10 σ12 + 3σ22

#

25 σ12 + 9 σ22

showing that Z1 , Z2 are, as a matter of fact, correlated. Example 3.6 Suppose that the coordinates x, y in a plane can be measured with uncertainties σ1 = 0.2 cm for the x-coordinate and σ2 = 0.4 cm for the y-coordinate. Assume further that the measured x, y values of a point in the

126 Probability theory plane are uncorrelated and they are considered the mean coordinates for that point. Our measurement yields (x, y) = (1, 1); what are the uncertainties in polar coordinates? Now, the functional relations for the problem are , r = x2 + y 2 θ = arctan(y/x) and the derivative matrix is * √ + * 1/√2 x/r −y/r2 = D= y/r x/r2 (x,y)=(1,1) 1/ 2

+

−1/2 1/2

while K = diag(σ12 , σ22 ) = diag(0.04, 0.16). Therefore * 0.100 T ∼ KD = K D = 0.042

+

0.042 0.050

√ and √ the uncertainties we are looking for are σr = 0.1 = 0.32 cm and σθ = 0.05 = 0.22 radians. Consequently, we will express our measurement as x = 1.0 ± 0.2; y = 1.0 ± 0.4 cm in rectangular coordinates and r = 1.41 ± 0.32 cm; θ = π/4 ± 0.22 radians in polar coordinates. Note that the transformation from rectangular to polar coordinates has introduced a positive correlation between r and θ.

3.6

Summary and comments

This chapter continues along the line of Chapter 2 by extending the discussion to the so-called multivariate case, that is, the case in which two, three, . . ., n random variable are considered simultaneously. In this light it is useful to introduce the concept of random vector and – whenever convenient – exploit the brevity and compactness of vector and matrix notation. A n-dimensional random vector X is, in essence, a measurable function from an abstract probability space (W, S, P) to Rn and this implies that each one of its components must be a random variable. In this light, Section 3.2 shows that the familiar concepts of induced probability measure, PDF and pdf (when it exists) can be readily extended to these vector-values functions. A new aspect, which has no counterpart in the one-dimensional case, is considered in Section 3.2.1 where the notion of marginal distribution functions is introduced. These functions have to do with the ‘subvectors’ of a given vector X and it is shown that the joint probability description of X contains implicitly the probabilistic description of each one of its possible ‘subvectors’. In general, however, the reverse statement is not true unless its components are independent. In this case, in fact, a number of important ‘product rules’ hold and one can obtain the joint-PDF (or pdf) of the vector from the PDFs (pdfs) of its components.

The multivariate case

127

Similarly to the one-dimensional case, the moments of a random vector are deﬁned as abstract Lebesgue integrals in the probability space (W, S, P). The most important moments in applications are the ﬁrst- and second-order moments which are given special names. So, in addition to the concepts of mean values and variances of X, the notion of covariance is deﬁned in Section 3.3 and some properties of these numerical descriptors are given. Particularly important in both theory and applications is the notion of uncorrelation of random variables which, broadly speaking, is a weak form of (pairwise) independence. Stochastic independence, in fact, implies uncorrelation but the reverse, in general, is not true. Besides this, Section 3.3 introduces the concept of joint-characteristic function by generalizing the one-dimensional case of Chapter 2; in particular, it is shown that independence implies the validity of a ‘product rule’ also for characteristic functions. Then, in Section 3.3.1 the discussion continues by noting the usefulness of matrix notation and by considering the actual calculations of moments and expectations in practice. In fact, mathematical analysis provides all the necessary results to show that the abstract Lebesgue integrals with respect to the measure P are evaluated as Lebesgue–Sieltjes integrals in Rn ; these, in turn, in most practical cases become either sums or ordinary Lebesgue integrals depending on the type of PDF – that is, FX – induced by the random vector X. Moreover, when the pdf exists the Lebesgue integrals coincide with the familiar Riemann integrals (it should be remembered, however, that Lebesgue integrals have a number of desirable properties which are not satisﬁed by Riemann integrals). Next, Section 3.3.2 is more application-oriented and gives two important examples of multivariate distributions: a discrete one, the so-called multinomial distribution, and a continuous one, the multivariate Gaussian (or normal) distribution. This is done in order to show how the developments considered so far are translated into practice. For its importance in both theory and practice, Sections 3.4 and 3.4.1 return on the subject of conditional probability. Here we extend the notion of conditioning to random variables by also considering, in the continuous case, the possibility of conditioning on events of zero probability. Then, since a conditional probability is a probability measure in its own right, the concepts of conditional PDF and pdf are introduced in the multivariate case and their relation to the joint and marginal functions is also shown. As one might expect, conditional expectations satisfy all the main properties of expectations. However, some additional properties are worthy of mention and these are given in Section 3.4.1 together with further theoretical remarks and examples. Finally, the last two Sections 3.5 and 3.5.1, deal with the probabilistic description of functions of a given random vector X, assuming that some information on X is available. More speciﬁcally – limiting for the most part the discussion to the continuous case – Section 3.5 considers the general problem of obtaining the joint-pdf of a vector Z = g(X); then, in

128 Probability theory order to show practical cases, some examples are given. On the other hand, Section 3.5.1 addresses the problem of obtaining some information on Z without necessarily trying to describe it completely. The task is accomplished by calculating the lowest-order moments – typically means, variances and covariances – of Z only on the basis of the available information on X. In most cases one only arrives at approximate relations because linearization of the function g is often necessary. Nonetheless, this partial information – obtained, in addition, by means of approximate equations – is sufﬁcient and sufﬁciently accurate in a large number of practical situations.

References and further reading [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Brémaud, P., ‘An Introduction to Probabilistic Modeling’, Springer-Verlag, New York (1988). [3] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press, 19th printing (1999). [4] Dall’Aglio, G., ‘Calcolo delle Probabilità’, Zanichelli, Bologna (2000). [5] Friedman, A., ‘Foundations of Modern Analysis’, Dover Publications, New York (1982). [6] Gnedenko, B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [7] Heathcote, C.R., ‘Probability, Elements of the Mathematical Theory’, Dover Publications, New York (2000). [8] Horn, R.A., Johnson, C.R., ‘Matrix Analysis’, Cambridge University Press (1985). [9] Kolmogorov, A.N., Fomin, S.V., ‘Introductory Real Analysis’, Dover, New York (1975). [10] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [11] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000). [12] Pfeiffer, P.E., ‘Concepts of Probability Theory’, 2nd edn., Dover Publications, New York (1978). [13] Rotondi, A., Pedroni, P., Pievatolo, A., ‘Probabilità, Statistica e Simulazione’, Springer-Verlag, Italia, Milano (2001). [14] Biswas, S., ‘Topics in Statistical Methodology’, Wiley Eastern Limited, New Delhi (1991). [15] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997). [16] Thompson, R.S.H.G., ‘Matrices: Their Meaning and Manipulation’, The English Univerities Press Ltd., London (1969). [17] Ventsel, E.S., ‘Teoria delle Probabilità’, Mir Publisher, Moscow (1983).

4

4.1

Convergences, limit theorems and the law of large numbers

Introduction

In most issues where chance plays a part, things seem to behave rather erratically if one looks only at a few instances. On the other hand, this type of behaviour seems to ‘smooth out’ in the long run. In other words, as the number of observed instances – or trials or experiments – increases, a more and more orderly pattern seems to ensue and certain regularities become clearer and clearer. This is what happens, for example, when we toss a coin; after 10 tosses we would not be surprised to have, say, eight heads and two tails but we would surely be if we got 800 heads and 200 tails after 1000 tosses. In fact, in this case we would seriously suspect that the coin is biased. This state of affair would be intriguing but not particularly interesting if it applied only to coins and dice. As a matter of fact, however, a large number of experiences in many ﬁelds of human activities – from birth and death rates to accidents, from measurements in science and technology to the occurrence of hurricanes or earthquakes, just to name a few – behave in a similar manner when measured, tabulated and/or assigned numerical values. The appearance of long-term regularities as the number of trials increases has been known for centuries and goes under the name of ‘law of large numbers’. The great achievement of probability theory is in having established the general conditions under which these regularities can and do occur. We open here a short parenthesis. Returning to the coin example for a moment, it is worth pointing out that the law of large numbers does not justify certain mistaken beliefs such as, say: I tossed a fair coin 15 times and I got 14 heads, the next toss is very likely to result in a head. This is wrong because the process has no memory and the probability of a head is 0.50 for each toss. In other words, the coin has no responsibility whatsoever to ‘make up’ for a past run of many heads in a row. This misinterpretation (unfortunately, a rather common misinterpretation; consider, for example, the habit of betting on ‘late’ numbers in lotteries) of the law of large numbers is due to the fact that one fails to distinguish between a regularity ‘in the ratio sense’ and a regularity in an ‘absolute sense’. The former concept refers to

130 Probability theory the number of heads (or tails) divided by the total number of tosses while the latter refers to the number of heads (or tails) in excess over tails (heads); as the number N of tosses increases, the above ratio tends to stabilize by getting closer and closer to 0.50 while the difference between heads and tails can become rather large (in fact, it generally increases). So, returning to our main discussion, this chapter is intended to provide the mathematical rationale behind the general term ‘law of large numbers’ and since the concept implies a tendency towards something, it is easily guessed that its mathematical formalization entails some kind of limit. The ﬁrst step, therefore, is to consider which kind of limits are involved in the long-term behaviour of experiments governed by chance.

4.2

Weak convergence

In the ﬁnal part of Section 2.4 (Deﬁnition 2.5) we introduced the notion of weak convergence of random variables. This type of convergence is also known in probability theory as ‘convergence in distribution’ or ‘convergence in law’ to mean that the probability law (i.e. the PDF) of Xn converges to a function which is itself a probability law. We recall here some important points: (a) Fn → F[w] – or equivalently Xn → X[D] – means that limn→∞ Fn (x) = F(x) at all points where F(x) is continuous (there is no ambiguity because F(x), being a PDF, is right-continuous). Also, it is not difﬁcult to see that Deﬁnition 2.5 of weak convergence is equivalent to stating that limn→∞ P(Xn ≤ x) = P(X ≤ x) whenever P(X = x) = 0; (b) since weak convergence does not refer directly to the r.v.s Xn and neither it involves directly the probability space on which they are deﬁned (weak convergence is a property of the PDFs and not of the Xn themselves), the concept makes sense even if the Xn are deﬁned on different probability spaces; (c) sequences of discrete r.v.s may converge (weakly) to a continuous r.v.s and conversely. Moreover, the fact that a sequence Xn of absolutely continuous r.v.s with pdfs fn = Fn converges in distribution to an absolutely continuous r.v. X whose pdf is f = F does not imply, in general, that the sequence fn converges to f . It is worth noting, however, that if fn → f pointwise (or even almost everywhere, see Section 4.3), then Xn → X[D]. The extension to random vectors is rather straightforward: if (Xn(1) , Xn(2) , . . . , Xn(m) ) converges weakly to the vector (X (1) , X (2) , . . . , X (m) ) then Xn(i) → X (i) [D] for every i = 1, 2, . . . , m. The reverse in general is not true and weak convergence of every individual component does not imply the vector weak convergence. This result should be hardly surprising; in fact, given F (1) and F (2) – we are considering the two-dimensional case for

Limits, convergences and the law of large numbers

131

F (1) , F (2)

simplicity – there are inﬁnite joint-PDFs for which are the marginal (i) (i) PDFs and therefore Fn → F [w] for i = 1, 2 gives no information on the convergence of FXn . In addition, FXn may not even converge at all. Also – in the light of the deﬁnition of weak convergence – it should be noted that Xn → X[D] does not imply Xn − X → 0[D], as it is customary for ordinary convergence of real variables. A fundamental result on D-convergence is given by Levy’s theorem of Proposition 2.24 which brings into play pointwise convergence of characteristic functions and is often used in probability theory. We use it, for instance, to prove a ﬁrst limit theorem: Proposition 4.1 Let Xn be a sequence of binomial r.v.s with parameters n and p = λ/n, where λ is a positive real number. Then, as n → ∞, Xn converges in distribution to a Poisson r.v. of parameter λ. Before proving this proposition, some preliminary comments on the Poisson distribution are in order. As it is probably known to the reader, we call Poisson r.v. with parameter λ a discrete r.v. X whose pmf is given by pX (x) = e−λ

λx x!

(x = 0, 1, 2, . . .)

(4.1)

and it can be shown that E(X) = Var(X) = λ. In fact, for example, E(X) =

∞ x=0

xe−λ

∞

λx−1 λx e−λ = λ =λ (x − 1)! x!

(4.2a)

x=1

because on the r.h.s. we sum on all the ordinates of the distribution and therefore the sum equals 1. In addition, the CF of the Poisson distribution is easily obtained as ϕ(u) = E(eiuX ) = e−λ

(λeiu )x x

−λ

=e

iu

x!

(4.2b) iu

exp(λe ) = exp[λ(e − 1)]

from which, using eqs (2.47b) and (2.34), it is almost immediate to determine that E(X 2 ) = λ + λ2 and Var(X) = λ. For higher-order moments it may be more convenient to use the recursion relation

d E(X k ) = λ + 1 E(X k−1 ) (4.2c) dλ with the starting assumption E(X 0 ) = 1. Therefore E(X) = λ, E(X 2 ) = λ + λ2 , E(X 3 ) = λ + 3λ2 + λ3 , E(X 4 ) = λ + 7λ2 + 6λ3 + λ4 , etc.

132 Probability theory A ﬁnal remark on the Poisson distribution is as follows: let X, Y be two independent Poisson r.v.s with parameters λ1 , λ2 , respectively. Independence implies (eq. (3.29)) that the CF of the r.v. X + Y is ϕX+Y (u) = {exp[λ1 (eiu − 1)]}{exp[λ2 (eiu − 1)]} = exp[(λ1 + λ2 )(eiu − 1)]

(4.3)

which is the CF of a Poisson r.v. with parameter λ1 + λ2 . This property of reproducing itself by addition of independent variables – possessed also by the Gaussian distribution – is noteworthy and often useful in practice. Moreover, a result by Rajkov shows that the reverse is also true: if the sum of two independent r.v. has a Poisson distribution then each individual r.v. is Poisson distributed. This, we recall (remark in Example 3.3) is true also for Gaussian r.v.s. Now, returning to our main discussion, we know from eq. (2.51) that the CF of the binomial r.v. Xn is given by ϕn (u) = (1 − λ/n + λeiu /n)n . Passing to the limit as n → ∞ we get lim ϕn (u) = lim

n→∞

n→∞

λ(eiu − 1) 1+ n

n = exp[λ(eiu − 1)]

(4.4)

which proves the assertion of Proposition 4.1. On the practical side, this proposition is interpreted by saying that the Poisson distribution – besides being often applicable in its own right – can be used as a valid approximation of the binomial distribution when the probability of ‘success’ p is rather small and n is sufﬁciently large. In fact it should be noted that all the binomial r.v.s Xn have the same mean E(Xn ) = pn = (λ/n)n = λ, thus implying that for large values of n the probability p must be small (incidentally, it is for this reason that the Poisson distribution is often called the distribution of rare events). In this light, as Example 4.1 will show, the parameter λ represents the average number of occurrences of the event under study per measurement unit (of time, length, area, etc., depending on the case). As a general rule of thumb one can use the Poisson distribution to approximate the binomial when either n ≥ 20 and p ≤ 0.05 or when n ≥ 100 and np ≤ 10; this makes calculations much easier because if we are interested in, say, the probability of 9 successes out of n = 1000 trials in a binomial process with p = 0.006 (so that λ = np = 6) it is certainly easier to calculate (69 e−6 )/9! rather than

1000 (0.006)9 (1 − 0.006)1000−9 9

(incidentally, the result of both expressions is 0.0688).

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Example 4.1 Two typical cases of Poisson r.v. are as follows. Consider the number of car accidents per month at a given intersection where it is known that, on average, there are 1.7 accidents per month. In this case the month is our measurement unit and the Poisson law can be justiﬁed as follows. Divide a month in n intervals, each of which is so small that at most one accident can occur with a probability p = 0. Then, since it is reasonable to assume that the occurrence of accidents is independent from interval to interval, we are in essence observing a Bernoulli trial where the probability of ‘success’ p is relatively small if n is large. Also we know that λ = np = 1.7 and we can, for instance, obtain the probability of zero accidents in a month as (1.70 e−1.7 )/0! = 0.183. The second example arises from a ballistic problem rather common during II World War. The probability of hitting an airplane in a vulnerable part when shooting with a riﬂe – that is, a ‘success’ – is very low, say, p = 0.001. However, if an entire military unit shoots, say, n = 4000 bullets, one can use the Poisson distribution to determine that the probability of at least two hits x −4 0 −4 1 −4 is (since λ = np = 4) 4000 x=2 4 e /x! = 1−(4 e /0!)−(4 e /1!) = 0.908 which is rather high and has been conﬁrmed in practice. Another important limit theorem – which involves D-convergence and points in the direction of the central limit theorem to be considered in a later section – was ﬁrst partially obtained by deMoivre in the eighteenth century and then completed by Laplace some 60–70 years later. Once again, one considers a sequence of Bernoulli trials and deﬁnes the random variables Xn (n = 1, 2, . . .) which take on the value 0 in case of ‘failure’ or the value 1 in case of ‘success’ (recall that the probability of ‘success’ p does not change from trial to trial). In this light the r.v. Sn = X1 + X2 + · · · + Xn represents the number of successes in n trials and is binomially distributed with mean √ np and standard deviation npq (Example 2.8a). With these assumptions we have the deMoivre–Laplace theorem: Proposition 4.2 Let Sn be the number of successes in a sequence of Bernoulli trials, then

b 1 Sn − np ≤b = √ exp(−z2 /2) dz lim P a < √ n→∞ npq 2π

(4.5)

a

uniformly for all a, b (−∞ ≤ a < b ≤ ∞). The proof is not given here because this proposition is just a particular case of the central limit theorem which will be proven in a later section (Proposition 4.22). Noting that the r.h.s. of eq. (4.5) is P(a ≤ Z < b) where Z is a standard Gaussian r.v., we can state Proposition 4.2 in words by saying that √ the sequence of r.v.s Yn = (Sn − np)/ npq – which, in turn, is obtained by

134 Probability theory ‘standardizing’ the sequence of binomial r.v.s Sn – converges in distribution to a standard Gaussian r.v. This result is also frequently expressed by saying that the r.v. Yn is ‘asymptotically standard normal’ and sometimes written Yn ≈ As−N(0, 1) where N(0, 1) denotes the normal probability distribution with zero mean and unit variance (i.e. the standard Gaussian distribution). In the light of the considerations above, it turns out that – in the limit of large n – the binomial distribution can be approximated either by a Poisson distribution or by a standardized Gaussian. Which one of the two approximations to use depends on the problem at hand; broadly speaking, the Gaussian approximation works well even for moderately large values of n (say n ≥ 20 − 25) as long as p is not too close to 0 or 1. If, on the other hand, p is close to 0 or 1, n must be rather large in order to obtain reasonably good results and in these cases the Poisson approximation is preferred. General rules of thumb are often given in textbooks and one ﬁnds, for example, that √ the Gaussian approximation is appropriate whenever (i) p ± 2 pq/n lies in the interval (0, 1) or (ii) np ≥ 5 if p ≤ 0.5 or nq ≥ 5 if p > 0.5. A third important and useful result considers the asymptotic behaviour of Poisson r.v.s. The CF of a Poisson r.v. X is given by eq. (4.2b);√as a consequence the CF of the standardized Poisson r.v. Y = (X − λ)/ λ is given by √ √ ϕY (u) = exp[−iu λ + λ(eiu/ λ − 1)]

(4.6)

where eq. (4.6) – since Y and X are linearly related – is obtained by using eq. (3.39b). ∞ we can expand the exponential in parenthesis as √ As λ → √ exp(iu/ λ) = 1 + iu/ λ − u2 /2λ + · · · and obtain lim ϕY (u) = exp(−u2 /2)

λ→∞

(4.7)

which, in other words, means that Y ≈ As − N(0, 1). In the light of Propositions 4.1 and 4.2, this last result is hardly unexpected. 4.2.1

A few further remarks on weak convergence

It has been pointed out in the preceding section that weak convergence (or convergence in distribution or in law) concerns the convergence of PDFs and, in general, does not imply the convergence of pmfs or pdfs (when they exist). However, in some cases there is the possibility of establishing ‘local’ limit theorems for these functions. An example is given by the ‘local’ version of the DeMoivre–Laplace theorem (see e.g. [9] or [13]) stating that √ npqBn (m) lim √ =1 n→∞ ( 2π )−1 exp(−x2 /2)

(4.8a)

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135

where

Bn (m) =

n m n m n−m p (1 − p)n−m = p q m m

(4.8b)

,

x = (m − np)/ npq

In words, the result of eq. (4.8a) is expressed by saying that, for any √ given m, the binomial pmf (multiplied by its standard deviation npq) tends to a standardized Gaussian pdf as n gets larger and larger. As a matter of fact, the approximation is rather good even for relatively small values of n. So, for example, if n = 25, p = 0.2 and we are inter√ ested in m = 3, √ then npqB25 (3) = 0.2715 and since x = −1 in this case, we get ( 2π )−1 exp(−x2 /2) = 0.2420. A graphical representation of this local theorem is given in Figures 4.1 (n = 25, p = 0.2) and 4.2 (n = 100, p = 0.2) where one can immediately notice the quality of the approximation: good in the ﬁrst case and excellent in the second case. The reader should check, however, that larger and larger values of n are needed for a good approximation as p gets close to either 0 or 1. As stated in the preceding section, when p is close to either 0 or 1 (say p < 0.1 or p > 0.9) the binomial pdf can be better approximated by a Poisson density. In fact, if we let n → ∞ and p → 0 so that λ = pn is ﬁnite,

0.42 Binomial Std. Gauss.

0.35 0.28 0.21 0.14 0.07 0.00 –2.5

–1.50

–0.50

0.50

1.50 x values

2.50

Figure 4.1 Gaussian approx. to binomial (n = 25, p = 0.2).

3.50

4.5

136 Probability theory 0.42 Binomial Std. Gauss.

0.35 0.28 0.21 0.14 0.07 0.00 –3.50

–2.50

–1.50

–0.50

0.50

1.50

2.50

3.50

x values

Figure 4.2 Gaussian approx. to binomial (n = 100, p = 0.2).

then for any ﬁxed value of m Bn (m) =

n! pm (1 − p)n−m m!(n − m)!

λm λ n−m 1 − m!nm n

m (1 − λ/n)n 1 λ m−1 = nm 1 − ··· 1 − n n m!nm (1 − λ/n)m = n(n − 1) · · · (n − m + 1)

and therefore, since (1 − λ/n)n → exp(−λ) as n → ∞, lim Bn (m) =

n→∞

λm −λ e m!

(4.9)

The approximations considered here, clearly, are not the only ones. So, for example, it may be reasonable to expect that a distribution arising from an experiment of sampling without replacement can be approximated by a distribution of a similar experiment with replacement if the total number of objects N from which the sample is taken is very large. In fact, as N → ∞ and we extract a ﬁnite sample, it no longer matters whether the extraction is done with or without replacement because the probability of ‘success’ is unaffected by the fact that we replace – or do not replace – the extracted item. In mathematical terms these considerations can be expressed by saying that, under certain circumstances, the so-called hypergeometric distribution – which is relative to sampling without replacement – can be approximated by a binomial distribution (see e.g. [18, Section 3.1.3] or [7, Appendix 1,

Limits, convergences and the law of large numbers

137

Section 17]). However, we do not consider other cases here and, if needed, postpone any further consideration.

4.3

Other types of convergence

Consider a sequence of r.v.s Xn deﬁned on the same probability space (W, S, P). We say that Xn converges in probability to the r.v. X – also deﬁned on (W, S, P) – if for every ε > 0 we have lim P{w ∈ W : |Xn (w) − X(w)| ≥ ε} = 0

n→∞

(4.10a)

and in this case one often writes Xn → X[P] and speaks of P-convergence. It is worth noting that convergence in probability is called ‘convergence in measure’ in mathematical analysis. In words, eq. (4.10a) states that the probability measure of the set where Xn differs from X by more than any prescribed positive number tends to zero as n → ∞. This, we point out, does not assure that all the values |Xn (w) − X(w)| will be smaller than ε for n larger than a certain N, but only that the probability measure of the event (i.e. set) for which |Xn (w) − X(w)| ≥ ε is very small (zero in the limit). Also, it may be noted that eq. (4.10) can be expressed equivalently by writing lim P{w ∈ W : |Xn (w) − X(w)| ≤ ε} = 1

n→∞

(4.10b)

and it is immediate to see that Xn → X[P] if and only if Xn − X → 0[P] (remember that this is not true in general for convergence in distribution). In the case of random vectors the condition (4.10a) – or (4.10b) – must hold for all their components and it is understood that the sequence of vectors Xn and the limit X must have the same dimension. More speciﬁcally, it can be shown that Xn → X[P] if and only if Xn(k) → X (k) [P] for all k (where k is here the index of component; so, for a m-dimensional vector k = 1, 2, . . . , m). We turn our attention now on some important results on convergence in probability starting with the following two propositions: Proposition 4.3 If Xn → X[P] and g : R → R is a continuous function, then g(Xn ) → g(X)[P]. Proposition 4.4(a) Convergence in probability implies convergence in distribution. In fact, we have Fn (x) = P(Xn ≤ x) = P(Xn ≤ x ∩ X > x + ε) + P(Xn ≤ x ∩ X ≤ x + ε) ≤ P(|X − Xn | ≥ ε) + P(X ≤ x + ε) = P(|X − Xn | ≥ ε) + F(x + ε)

138 Probability theory where the inequality comes from two facts: (a) P(Xn ≤ x ∩ X ≤ x + ε) ≤ P(X ≤ x + ε) because of the straightforward inclusion (Xn ≤ x ∩ X ≤ x + ε) ⊆ (X ≤ x + ε), and (b) P(Xn ≤ x ∩ X > x + ε) ≤ P(|X − Xn | ≥ ε) because (Xn ≤ x ∩ X > x + ε) ⊆ (|X − Xn | ≥ ε). This inclusion is less immediate but the l.h.s. event implies x < X − ε and, clearly, Xn ≤ x; consequently Xn < X − ε, which, in turn, is included in the event |X − Xn | ≥ ε. By a similar line of reasoning we get F(x − ε) = P(X ≤ x − ε) = P(X ≤ x − ε ∩ Xn > x) + P(X ≤ x − ε ∩ Xn ≤ x) ≤ P(|X − Xn | ≥ ε) + P(Xn ≤ x) = P(|X − Xn | ≥ ε) + Fn (x) Putting the two pieces together leads to F(x − ε) − P(|X − Xn | ≥ ε) ≤ Fn (x) ≤ P(|X − Xn | ≥ ε) + F(x + ε) and since Xn → X[P] then Fn (x) is bracketed between two quantities that – as ε → 0 – tend to F(x) whenever F is continuous at x. This, in turn, means that Xn → X[D] and the theorem is proven. The reverse statement of Proposition 4.4a is not true in general because – we recall – convergence in distribution can occur for r.v.s deﬁned on different probability spaces, a case in which P-convergence is not even deﬁned. However, when the Xn are deﬁned on the same probability space, a partial converse exists: Proposition 4.4(b) If Xn converges in distribution to a constant c then Xn converges in probability to c. We do not prove the proposition but only point out that: (i) a r.v. which takes on a constant value c with probability one – that is, such that PX (c) = 1 – is not truly random. Its PDF is F(x) = 0 for x < c F(x) = 1 for x ≥ c and often one speaks of ‘degenerate’ or ‘pseudo’ random variable in this case; (ii) when all the Xn and X are deﬁned on the same probability space and X is not a constant, there are special cases in which the converse of Proposition 4.4 may hold (see [11, Chapter 4]). The last result on P-convergence we give here is called Slutsky’s theorem and its proof can be found, for example, in Ref. [1]

Limits, convergences and the law of large numbers

139

If Xn → X [D] and Yn → c [D] (and therefore Yn →

Proposition 4.5 c [P]), then

(a) Xn + Yn → X + c [D] (b) Xn Yn → cX [D] (c) Xn /Yn → X/c [D] if c = 0. Turning now to another important notion of convergence, we say that the sequence of r.v.s Xn converges almost-surely (some authors say ‘with probability 1’) to X if 3 4 P w ∈ W : lim Xn (w) = X(w) = 1 n→∞

(4.11)

and we will write Xn → X [a.s.] or Xn → X[P − a.s.] if the measure needs to be speciﬁed. Clearly, Xn → X [a.s.] if and only if Xn − X → 0 [a.s.]. Deﬁnition 4.11 implies that the set N of all w where Xn (w) fails to converge to X(w) is such that P(N) = 0 and that, on the other hand, Xn (w) → X(w) for all w ∈ N c where, clearly, P(N c ) = 1. Given a measure P – and a probability is a ﬁnite, non-negative measure – in mathematical analysis one speaks of ‘convergence almost-everywhere’ (a.e.) when condition (4.11) holds; therefore a.s.-convergence is just the probabilistic name given to the notion of a.e.-convergence of advanced calculus. In general, there is no relation between a.e.-convergence and convergence in measure (eq. (4.10)); however, the fact that P is a ﬁnite measure has an important consequence for our purposes: Proposition 4.6 Almost-sure convergence implies convergence in probability (and therefore, by Proposition 4.4, convergence in distribution). This result is a consequence of the following criterion for a.s.-convergence: the sequence Xn converges almost surely to X if and only if for every ε > 0 ⎡ ⎤ ∞ (4.12a) lim P ⎣ {|Xk − X| ≥ ε}⎦ = 0 n→∞

k=n

or, equivalently, ⎡ lim P ⎣

n→∞

∞

⎤ {|Xk − X| < ε}⎦ = 1

(4.12b)

k=n

In fact, if (4.12a) holds then eq. (4.10) follows by virtue of the fact that the probability of a union of events is certainly not less than the probability of each one of the individual events in the union. The proof of the criterion is more involved and is not given here; the interested reader may refer, for

140 Probability theory example, to [16] or [17]. Regarding the converse of Proposition 4.6 – which is not, in general, true – a remark is worthy of notice: it can be shown that if Xn → X[P] then there exists a subsequence Xnk of Xn such that Xnk → X [a.s.] as k → ∞. Proposition 4.7 If Xn → X [a.s.] and g is a continuous function, then g(Xn ) → g(X) [a.s.]. In fact, for every ﬁxed w such that Xn (w) → X(w) then Yn (w) ≡ g(Xn (w)) → g(X(w)) ≡ Y(w) because of the continuity of g. Therefore {w: Xn (w) → X(w)} ⊆ {w: Yn (w) → Y(w)} so that P{w: Yn (w) → Y(w)} ≥ P{w: Xn (w) → X(w)} and the theorem follows. The last comment we make here on a.s.-convergence regards random vectors. As for P-convergence, a sequence of m-dimensional random vectors Xn = (Xn(1) , . . . , Xn(m) ) converges a.s. to the m-dimensional vector X = (X (1) , . . . , X (m) ) if and only if Xn(k) → X (k) [a.s.] for all k = 1, 2, . . . , m. Before turning to the collection of results known as ‘law of large numbers’, we close this section by introducing another type of convergence. A sequence of r.v.s Xn is said to converge to X ‘in the kth mean’ (k = 1, 2, . . .) if lim E(|Xn − X|k ) = lim |Xn − X|k dP = 0 (4.13) n→∞

n→∞

W

and we will write Xn → X [Mk ]. In the above deﬁnition it is assumed that all the Xn and X are such that E(Xnk ) < ∞ and E(X k ) < ∞ because these conditions imply the existence of the expectation in eq. (4.13). In fact, from the inequality |Xn − X|k ≤ 2k (|Xn |k + |X|k ) we can pass to expectations to get E(|Xn − X|k ) ≤ 2k E(|Xn |k ) + 2k E(|X|k ) so that the l.h.s. is ﬁnite whenever the r.h.s. is. Also, it is easy to see that Xn → X [Mk ] if and only if Xn − X → 0 [Mk ]. The most important special cases of (4.13) in applications are k = 1 – the so-called ‘convergence in the mean’ – and k = 2, called ‘convergence in the quadratic mean’. This latter type plays a role in probability when only ‘second-order data’ are available, that is, when the only information is given by the means mn = E(Xn ) and covariances Kij (i, j = 1, . . . , n) and one cannot determine whether the sequence converges in any one of the modes considered before. However, the following result holds: Proposition 4.8 If Xn → X[Mk ] – with k being any one integer – then Xn → X [P] and therefore (Proposition 4.4) Xn → X [D]. In fact, consider Chebyshev’s inequality (eq. (2.36a)) applied to the r.v. Xn − X; for every ε > 0 we have P(|Xn − X| ≥ ε) ≤ E(|Xn − X|k )/ε k and therefore the l.h.s. tends to zero whenever the r.h.s. does. So, in particular, if a sequence converges in the mean or in the quadratic mean then convergence

Limits, convergences and the law of large numbers

141

in probability and convergence in distribution follow. Furthermore, by virtue of Proposition 2.12, it is immediate to show that convergence in the quadratic mean implies convergence in the mean or, more generally: Proposition 4.9 Convergence in the kth mean implies convergence in the jth mean for all integers j ≤ k. 4.3.1

Additional notes on convergences

In the preceding section we have determined the following relations: (a) a.s.-convergence is stronger than P-convergence which, in turn, is stronger than D-convergence unless the limit is a constant random variable. (b) Mk -convergence (for any one integer k) implies P-convergence and therefore D-convergence. At this point one may ask, for instance, about the relation between Mk and a.s.-convergence. The answer is that, in general, without additional assumptions, there are no relations other than the ones given above. An example is given by the celebrated Lebesgue dominated convergence theorem which, for our purposes, can be stated as follows Proposition 4.10 Let Xn → X [a.s.] or Xn → X [P] and let Y be a r.v. such that E(Y) < ∞ (i.e. with ﬁnite mean) and |Xn (w)| ≤ Y(w) for each n and for almost all w ∈ W. Then Xn → X [M1 ]. (see Ref. [8] or [15]). Note that the expression |Xn (w)| ≤ Y(w) for almost all w ∈ W brings into play the measure P and means that the set N where the inequality does not hold is such that P(N) = 0 (again, this is the ‘almost everywhere’ notion of mathematical analysis). Another important result establishes a relation between D- and a.s.convergence. This is due to Skorohod and, broadly speaking, states that convergence in distribution can be turned into almost sure convergence by appropriately changing probability space. Proposition 4.11 (Skorohod’s theorem) Let Xn and X be r.v.s deﬁned on a probability space (W, S, P) and such that Xn → X [D]. Then, it is possible 5-n and X to construct a probability space (W, S, P) and random variables X such that P(X ≤ x) = P(X ≤ x), P(Xn ≤ x) = P(Xn ≤ x) for n = 1, 2, . . . -n → X - [ˆ (i.e. F(x) = F(x) and Fˆ n (x) = Fn (x) for all n) and X P − a.s.]. We do not prove the theorem here but it is worth noting that, in essence, Proposition 4.11 is due to the fact that any PDF F : R → [0, 1] can be ‘inverted’ to obtain a r.v. deﬁned on the interval U = [0, 1] whose PDF

142 Probability theory 5is F. In this light, it turns out that (W, S, P) = (U, B(U), µ) – where µ is the Lebesgue measure. For more details the interested reader can refer, for example, to [1, 2] or [19]. A third remark of interest is that P-, a.s.- and Mk -convergence can all be established by the well-known Cauchy criterion of mathematical analysis. So, for example, if a sequence Xn satisﬁes the Cauchy criterion in probability, that is, lim P(|Xm − Xn | ≥ ε) = 0

m,n→∞

(4.14)

(which can also be written Xm − Xn → 0 [P] as m, n → ∞), then there exists a r.v. X such that Xn → X [P]. The fact that Xn → X [P] implies eq. (4.14) is clear; therefore it can be said that the Cauchy criterion (4.14) is a necessary and sufﬁcient condition for the sequence Xn to converge (in probability) to a r.v. X deﬁned on the same probability space. Similarly, it can be shown that |Xm − Xn | → 0 [a.s.] implies that there exists X such that Xn → X [a.s.]; consequently, by the same reasoning as above Xn → X [a.s.] if and only if |Xm − Xn | → 0 [a.s.]. By the same token, Xn → X [Mk ] if and only if the Cauchy criterion E(|Xm − Xn |k ) → 0(m, n → ∞) in the kth mean holds. In mathematical terminology, these results can be expressed by saying that the ‘space’ of random variables deﬁned on a probability space (W, S, P) is complete with respect to P, a.s. and Mk convergence. Moreover, if we consider as equal any two r.v.s which are almost everywhere equal (with respect to the measure P) the spaces of r.v.s. with ﬁnite kth order moment (k = 1, 2, . . .) are the so-called Lk spaces of functional analysis. It is well known, in fact, that deﬁning the norm Xk = {E(|X|k )}1/k these are Banach spaces (i.e. complete normed spaces) and, in particular, the space L2 is a Hilbert space. Although it is beyond our scopes, this aspect of probability theory has far-reaching consequences in the light of the fact that the study of Banach and Hilbert spaces is a vast and rich ﬁeld of mathematical analysis in its own right.

4.4

The weak law of large numbers (WLLN)

Broadly speaking, the so-called ‘law of large numbers’ (LLN) deals with the asymptotic behaviour of the arithmetic mean of a sequence of random variables. Since the term ‘asymptotic behaviour’ implies some kind of limit and therefore a notion of convergence, it is customary to distinguish between the ‘weak’ law of large numbers (WLLN) and ‘strong’ law of large numbers (SLLN), where in the former case the convergence is in the probability sense while in the latter almost sure convergence is involved. Clearly, the attributes of ‘weak’ and ‘strong’ are due to the fact that a.s.-convergence is stronger than P-convergence and therefore the SLLN implies the WLLN. In order to cast these ideas in mathematical form, let us consider the WLLN ﬁrst and start with a general result which is a consequence of Chebychev’s

Limits, convergences and the law of large numbers

143

inequality. For n = 1, 2, . . . consider a√ sequence {Yn } of r.v.s with ﬁnite means E(Yn ) and standard deviations σn = Var(Yn ). Then our ﬁrst statement is: Proposition 4.12 If the numerical sequence of standard deviations is such that σn → 0 as n → ∞, then for every ε > 0 lim P(|Yn − E(Yn )| ≥ ε) = 0

n→∞

(4.15)

By setting b = ε, the proof follows immediately from the ﬁrst of eq. (2.36b). Now, given a sequence of r.v.s Xk deﬁned on a probability space (W, S, P) we can deﬁne, for every n = 1, 2, . . ., the new r.v. Sn = X1 + X2 + . . . + Xn

(4.16)

n with mean E(Sn ) and variance Var(Sn ). (Note that E(Sn ) = k=1 E(Xk ) while, in the general case, eq. (2.35b) gives Var(Sn ) in terms of the variances and covariances of the original variables Xk . If these variables are independent or uncorrelated then Var(Sn ) = nk=1 Var(Xk ).) With these deﬁnitions in mind, the following propositions hold: Proposition 4.13 (Markov’s WLLN)

Sn − E(Sn ) ≥ε =0 lim P n→∞ n

If Var(Sn )/n2 → 0 as n → ∞ then (4.17)

Proposition 4.14(a) (Chebychev’s WLLN) If the variables Xk are independent or uncorrelated and there exists a ﬁnite, positive constant C such that Var(Xk ) < C for all k (in other words, this latter condition can be expressed by saying that the variances Var(Xk ) are ‘uniformly bounded’), then eq. (4.17) holds. The proof of Proposition 4.13 is almost immediate. If we set Yn = Sn /n then, by hypothesis, Var(Yn ) = Var(Sn )/n2 → 0 as n → ∞ and E(Yn ) = E(Sn )/n. In this light, Proposition 4.13 is a consequence of Proposition 4.12. For n Proposition 4.14 we note ﬁrst that Var(Sn ) = k=1 Var(Xk ) < nC, where the equality holds because of independence (or uncorrelation). Consequently, Var(Sn )/n2 < C/n, and since C/n → 0 as n → ∞ the result follows by virtue of Proposition 4.13. At this point, some remarks are in order. First of all, we note that eq. (4.17) can be rewritten equivalently as (Sn − E(Sn ))/n → 0[P] or Sn /n → E(Sn )/n[P], where Sn /n is the arithmetic mean of the r.v.s X1 , X2 , . . . , Xn . So, if the Xk are such that E(Xk ) = µ for all k, then E(Sn ) = nµ and Sn /n → µ [P]

(4.18)

144 Probability theory meaning that for large n the arithmetic mean of n independent r.v.s (each with ﬁnite expectation µ and with uniformly bounded variances) is very likely to be close to µ. This is what happens, for instance, when we repeat a given experiment a large number of times. In this case we ‘sample’ n times a given r.v. X – which is assumed to have ﬁnite mean E(X) = µ and variance Var(X) = σ 2 – so that X1 , X2 , . . . , Xn are independent r.v.s distributed as X. Then, by calculating the arithmetic mean (X1 + X2 + · · · + Xn )/n of our n observations we expect that X 1 + X 2 + · · · + Xn ∼ Sn = =µ n n

(4.19)

We will have more to say about this in future chapters but, for the moment, we note that a typical example in this regard is the measuring process of an unknown physical quantity Q: we make n independent measurements of the quantity, calculate the mean of these observed values and take the result as a good (if n is sufﬁciently large) estimate of the ‘true value’ Q. Note that the assumptions of Proposition 4.14 are satisﬁed because all the Xk have the same distribution as X so that, in particular, E(Xk ) = µ (if the measurements have no systematic error) and Var(Xk ) = σ 2 (and since σ is a ﬁnite number, the variances are uniformly bounded). The relative frequency interpretation of the probability of an event A (recall Section 1.3) is also dependent on the LLN. In fact, by performing n times an experiment in which A can occur, the relative frequency f (A) of A is 1 f (A) = Ik n n

(4.20)

k=1

where Ik is the indicator function of event A in the kth repetition of the experiment. As n gets larger and larger, it is observed that f (A) tends to stabilize in the vicinity of a value – for example, 0.50 in the tossing of a fair coin or, say, 0.03 for the fraction of defective items in the daily production of a given industrial process – which, in turn, is postulated to be the probability of A. In this light, it is clear that we cannot rigorously prove or disprove the existence, in the real world, of such a limiting value because an inﬁnite number of trials is impossible. The best we can do is to build up conﬁdence in our assumptions and check them against real observations; continued success tends to increase our conﬁdence, thus leading us to believe in the adequacy of the postulate. Returning to our main discussion we note that a special case of Proposition 4.14 is given by the celebrated Bernoulli theorem whose basic assumption is that we perform a sequence of Bernoulli trials and p is the probability of success in each trial. If Xk = Ik – the indicator function of a success in the kth trial – the sum Sn is the total number of successes in

Limits, convergences and the law of large numbers

145

n trials and is binomially distributed with (Example 2.8) E(Sn ) = np and Var(Sn ) = np(1 − p) = npq. Then, Bernoulli’s theorem asserts: Proposition 4.14(b) (Bernoulli’s WLLN)

Sn lim P − p < ε = 1 n→∞ n

With the above assumptions (4.21)

or, equivalently, Sn /n → p[P]. The proof follows from Markov’s theorem (Proposition 4.13) once we note that Var(Sn )/n2 = pq/n → 0 as n → ∞. Now, although the proof of the theorem may seem almost trivial, we must keep in mind that it was the ﬁrst limit theorem to be proved (in the book Ars Conjectandi published in 1713), and therefore Bernoulli did not have the mathematical resources at our disposal. Moreover, since the theorem states that the average number of successes in a long sequence of trials is close to the probability of success on any given trial, its historical importance lies in the fact that this is the ﬁrst step in the direction of removing the restriction of ‘equally likely outcomes’ – necessary in the ‘classical’ notion of probability – in deﬁning the probability of an event. As a consequence, it provides mathematical support to the idea that probabilities can be determined as relative frequencies in a sufﬁciently long sequence of repeated trials. The different forms of the WLLN given so far assume that all the variables Xi have ﬁnite variance. Khintchine’s theorem shows that this is not necessary if the variables are independent and have the same distribution. Proposition 4.15 (Khintchine’s WLLN) If the r.v.s Xk are independent and identically distributed (iid) with ﬁnite ﬁrst moment E(Xk ) = µ then Sn /n → µ [P]. In order to prove the theorem we can use characteristic functions to show that Sn /n → µ[D]. This, by virtue of Proposition 4.4(b), implies convergence in probability. Let ϕ(u) be the common CF of the variables Xk , then we can write the MacLaurin expansion ϕ(u) = ϕ(0) + iuE(Xk ) + · · · = 1 + iuµ + · · · (see Proposition 2.18(a) and the ﬁrst of eq. (2.47b)) where the excluded terms tend to zero as u → 0. Then, if we call ψ(u) = E[exp(iuSn )] the CF of Sn we have E[exp(iuSn /n)] = ψ(u/n) =

n

ϕ(u/n) = {ϕ(u/n)}n = (1 + iuµ/n + · · · )n

k=1

where we used independence in the second equality. Now, as n → ∞, the last expression on the r.h.s. tends to exp(iuµ) which, in turn, is the CF of a pseudo-r.v. µ. Consequently, Sn /n → µ [D] and therefore

146 Probability theory Sn /n → µ[P]. A different proof of this theorem is based on the so-called ‘method of truncation’ and can be found, for example, in [2] or [9]. At this point it could be asked if there is a necessary and sufﬁcient condition for the WLLN to hold. In fact, all the results above provide sufﬁcient conditions and examples can be given of sequences which obey the WLLN but do not verify the assumptions of any one of the theorems above. Such a condition exists and is given in the next theorem due to Kolmogorov. Proposition 4.16 (Kolmogorov’s WLLN) A sequence Xk of r.v.s with ﬁnite expectations E(Xk ) satisﬁes eq. (4.17) – that is, the WLLN – if and only if

2n lim E n→∞ 1 + 2n

6 =0

(4.22)

where = [Sn − E(Sn )]/n. We do not prove this proposition here and the interested reader may refer, for example, to [9]. However, it is worth noting that the theorem requires neither independence nor the existence of ﬁnite second-order moments. Also, since Kolmogorov’s theorem expresses an ‘if and only if’ statement, it can be said that the various conditions of the propositions above are all sufﬁcient conditions for (4.22) to hold, meaning that they imply (but are not implied by) eq. (4.22). In fact, for example, in case of ﬁnite variances we have 2n 1 ≤ 2n = 2 (Sn − E(Sn ))2 2 1 + n n so that taking expectations on both sides it follows that (4.22) holds whenever Markov’s condition on variances (Proposition 4.13) holds.

4.5

The strong law of large numbers (SLLN)

As stated in the preceding section, the type of convergence involved in the different forms of the SLLN is a.s.-convergence, which, in turn, implies P- and D-convergence. Being a stronger statement than the WLLN, the mathematical proofs of the SLLN are generally longer and more intricate than in the weak case; for this reason we will mainly limit ourselves to the results. The reader interested in the proofs of the theorems can ﬁnd them in the references at the end of the chapter. Historically, the ﬁrst statement of SLLN is due to Borel and is somehow a stronger version of Bernoulli’s theorem (Proposition 4.15): Proposition 4.17 (Borel’s SLLN) Let Xk = Ik (k = 1, 2, . . .) be the indicator function of a success in the kth trial in a sequence of independent trials

Limits, convergences and the law of large numbers

147

and let p be the probability of success in each trial. Then Sn /n → p [a.s.], where, as before, Sn = X1 + X2 + · · · + Xn . Borel’s theorem, in turn, is a special case of the more general result due to Kolmogorov: Proposition 4.18 (Kolmogorov’s SLLN) Let Xk be a sequence of independent r.v.s with ﬁnite variances Var(Xk ) such that ∞ Var(Xk ) k=1

k2

0

P

max |Sk − E(Sk )| ≥ b ≤

1≤k≤n

Var(Sn ) b2

(4.25)

Note that if n = 1 eq. (4.25) reduces to Chebishev’s inequality (2.36b). Proposition 4.21

Borel–Cantelli lemma consists of two parts:

(a) Let (W, S, P) be a probability space and A1 , A2 , . . . be a sequence of events (i.e. An ∈ S for all n = 1, 2, . . .). If ∞ n=1 P(An ) < ∞, then

P lim sup An = 0

(4.26a)

n→∞

∞ where, we recall from Appendix A, lim supn→∞ An = ∩∞ n=1 ∪k=n Ak is itself an event which, by deﬁnition, occurs if and only if inﬁnitely many of the An s occur. (b) If A1 , A2 , . . . are mutually independent and ∞ n=1 P(An ) = ∞, then

P lim sup An = 1 n→∞

(4.26b)

Limits, convergences and the law of large numbers

149

∪∞ A . k=n k

To prove the ﬁrst part of the lemma, set En = Then E1 ⊃ E2 ⊃ E3 ⊃ · · · is a decreasing sequence of events and the theorem follows from the chain of relations

∞ P lim sup An = P lim En = lim P(En ) ≤ lim P(Ak ) = 0 n→∞

n→∞

n→∞

n→∞

k=n

where we used the deﬁnition of limit of a decreasing sequence of sets (see Appendix A, Section A1) ﬁrst and then the continuity and subadditivity properties of probability. For the second part of the lemma we can write ⎛ P(EC n)

= P⎝

∞

⎞ ⎠ AC k

⎛ = lim P ⎝

k=n

= lim

m→∞

m

k=n

⎞ ⎠ AC k

k=n m

[1 − P(Ak )] ≤ lim

m→∞

⎡

= lim exp ⎣− m→∞

m→∞

m

m

⎤

= lim

m→∞

m

P(AC k)

k=n

exp[−P(Ak )]

k=n

P(Ak )⎦ = 0

k=n

where we used, in the order, De Morgan’s law (i.e. eq. A.6), the independence of the Ak and the inequality 1 + x ≤ ex (which holds for any real number x). Moreover, the last equality holds because ∞ k=n P(Ak ) = ∞. From the relations above, part (b) of the lemma follows from the fact that P(En ) = 1 − P(EC n ) = 1 for each n.

4.6

The central limit theorem

In problems where probabilistic concepts play a part it is often reasonable to assume that the unpredictability may be due to the overall effect of many random factors and that each one of them has only a small inﬂuence on the ﬁnal result. Moreover, these factors – being ascribable to distinct and logically unrelated causes – can be frequently considered as mutually independent. Since our interest lies in the ﬁnal result and not in the individual factors themselves – which, often, are difﬁcult or even impossible to identify – it becomes important to study the existence of limiting probability distributions when an indeﬁnitely large number of independent random effects combine to yield an observable outcome. Needless to say, the ubiquitous Gaussian distribution is one of these limits and the mathematical results formalizing this fact – that is, convergence to a Gaussian distribution – go under the general name of ‘central limit theorem’ (CLT). The various forms of the theorem differ in the assumptions made on the probabilistic nature of the causes affecting the ﬁnal result.

150 Probability theory In order to cast the above ideas in mathematical form, we consider a sequence X1 , X2 , . . . of independent random variables with ﬁnite means E(X1 ) = m1 , E(X2 ) = m2 , . . . and variances Var(X1 ) = σ12 , Var(X2 ) = σ22 , . . .; also, as in the previous sections, we denote by Sn the sum Sn = X1 + X2 + · · · + Xn whose mean and variance are E(Sn ) = nk=1 mk and Var(Sn ) = nk=1 σk2 , respectively. The simplest case is when the variables Xk are iid; then, by calling m ≡ m1 = m2 = · · · the common mean and σ 2 ≡ σ12 = σ22 = · · · the common variance, we have E(Sn ) = nm and Var(Sn ) = nσ 2 . Since these quantities both diverge as n → ∞, it cannot be expected that the sequence Sn can converge in distribution to a random variable with ﬁnite mean and variance (unless, of course, the case m = 0 and σ = 0). However, we can turn our attention to the ‘standardized’ sequence Yn deﬁned by Sn − E(Sn ) Sn − nm Yn = √ = √ σ n Var(Sn )

(4.27)

whose mean and variance are, respectively, 0 and 1. In this light, the ﬁrst form of the CLT – also known as Lindeberg–Levy theorem – is as follows: Proposition 4.22 (Lindeberg–Levy: CLT for iid variables) Let Xk (k = 1, 2, . . .) be iid random variables with ﬁnite mean m and variance σ 2 ; then Yn → Z[D] as n → ∞, where the symbol Z denotes the standardized Gaussian r.v. whose PDF is 1 FZ (x) = √ 2π

x exp(−t 2 /2) dt −∞

The proposition can be proven by recalling Levy’s theorem (Proposition 2.24) and using characteristic functions. In fact, by introducing the iid r.v.s (with zero mean and unit variance) Uj = (Xj − m)/σ for j = 1, 2, . . ., we have Sn − nm 1 Uj =√ √ σ n n n

Yn =

j=1

Now, denoting by ϕ(u) the common CF of the variables Uj , the existence of ﬁnite mean and variance allows one to write the MacLaurin expansion ϕ(u) = ϕ(0) + uϕ (0) +

u2 u2 ϕ (0) + · · · = 1 − + ··· 2 2

where the dots indicate higher order terms that tend to zero more rapidly than u2 as u → 0. Then, since by virtue of independence the CF ψn of Yn is

Limits, convergences and the law of large numbers √ given by ψn (u) = {ϕ(u/ n)}n , we have √

u2 ψn (u) = {ϕ(u/ n)} = 1 − + ··· 2n

151

n

n

so that letting n → ∞ we get limn→∞ ψn (u) = exp(−u2 /2) (the technicality of justifying the fact that we neglect higher order terms can be tackled by passing to natural logarithms; for more details the reader can refer to [19, Chapter VI, Section 7]). This limiting function is precisely the CF of a standardized Gaussian r.v., therefore proving the assertion Yn → Z ≈ N(0, 1)[D] which, more explicitly, can also be expressed as 1 lim P(a < Yn ≤ b) = √ n→∞ 2π

b exp(−x2 /2) dx

(4.28a)

a

for all a, b such that −∞ ≤ a < b ≤ ∞. Equivalently, by taking a = −1 and b = 1 we can also write

9 1 Sn 2 σ lim P − m < √ exp(−x2 /2) dx = n→∞ n π n

(4.28b)

0

which, for large n, can also be interpreted as an estimate on the probability that the arithmetic mean Sn /n (see √ also the following remark (c)) takes values within an interval of length 2σ/ n centered about the mean m. At this point, a few remarks are in order: (a) The DeMoivre–Laplace theorem (Proposition 4.2) is a special case of CLT of Proposition 4.22. In DeMoivre–Laplace case, in fact, the variables Xj are all binomially distributed with mean p and variance pq = p(1 − q). Consequently, the mean and variance of Sn – the number of successes in n independent trials – are np and npq, respectively, so that eq. (4.28) reduces to eq. (4.5). (b) If the Xj are (independent) Poisson r.v.s with parameter λ, then – by virtue of the ‘self-reproducing property’ of Poisson variables pointed out in Section 4.2 – the variable Sn is also Poisson distributed with parameter = nλ and the CF ψn of the variable Yn is obtained by simply substituting in place of λ in eq. (4.6). Then ψn (u) → exp(−u2 /2) as n → ∞, therefore leading to another important special case of Proposition 4.22. (c) In different words, the statement of Proposition 4.22 can be expressed by saying that the variable Sn is asymptotically Gaussian with mean nm

152 Probability theory and variance nσ 2 . This, in turn, implies that the arithmetic mean 1 Sn Xk = n n n

Xn =

(4.29)

k=1

is itself a r.v. which is asymptotically normal with)mean E(X n ) = m, √ variance Var(X n ) = σ 2 /n and standard deviation Var(X n ) = σ/ n. This fact, we will see, often plays an important role in cases where a large number of elements is involved. In particular, it is at the basis of Gauss’ theory of errors where the experimental value of the (unknown) quantity, say Q, is ‘estimated’ by calculating the arithmetic mean of many repeated measurements under the assumption that the errors are iid random variables with zero mean and ﬁnite variance σ 2 . Then – besides relying on the SLLN stating that X n → Q [a.s.] – if n is sufﬁciently large we can also use the Gaussian distribution to make probability statements regarding the accuracy of our result. More about these and other statistical applications is delayed to later chapters. (d) Berry–Esseen inequality: Since for large values of n the standardized Gaussian can be considered as an approximation of the PDF of the variable Yn , the question may arise on how good is this estimate as a function of n. Now, besides the practical fact – also supported by the results of many computer simulations – that the approximation is generally rather good for n ≥ 10, a more deﬁnite answer can be obtained if one has some additional information on the X variables. If, for example, it is known that these variables have a ﬁnite third-order absolute central moment – that is, E(|X − m|3 ) < ∞ – a rather general result is given by Berry–Esseen inequality which states that for all x |Fn (x) − FZ (x)| ≤ C

E(|X − m|3 ) √ σ3 n

(4.30)

where we called Fn (x) = P(Yn ≤ x) the PDF of Yn , FZ (x), as above, is the standardized Gaussian PDF and C is a constant whose current best estimate is C = 0.798 (see Refs [11–14]). Although Lindeberg–Levy theorem is important and often useful, the requirement of iid random variables is too strict to justify all the cases in which the Gaussian approximation seems to apply. In fact, other forms of the CLT show that the assumption of identically distributed variables can be relaxed without precluding the convergence to the Gaussian distribution. Retaining the assumption of independence, a classical result in this direction is Lindeberg’s theorem. We state it without proof and the interested reader can refer, for instance, to [1, 9] or [19].

Limits, convergences and the law of large numbers

153

Proposition 4.23 (Lindeberg’s CLT) Let X1 , X2 , . . . be a sequence of independent random variables with ﬁnite means E(Xk ) = mk and variances Var(Xk ) = σk2 (k = 1, 2, . . .) and let Fk (x) be the PDF of Xk . If, for every ε > 0 (ε enters in the domain of integration, see eq. (4.31b)) the Lindeberg condition 1 n→∞ Var(Sn ) n

lim

(x − mk )2 dFk (x) = 0

(4.31a)

k=1C k

holds, √ then Yn → Z ≈ N(0, 1)[D], that is, the variable Yn = [Sn − E(Sn )]/ Var(Sn ) converges in distribution to a standardized Gaussian r.v. Z. Two remarks on notation: √ n n 2 (i) Clearly E(Sn ) = Var(Sn ) is the k=1 mk , Var(Sn ) = k=1 σk and standard deviation of Sn . In the following, for brevity these last two parameters will often be denoted by Vn2 and Vn , respectively. (ii) The domain of integration Ck in condition (4.31) is the set deﬁned by Ck = {x : |x − mk | ≥ εVn }

(4.31b)

Basically, the Lindeberg condition is an elaborate – and perhaps rather intimidating-looking – way of requiring that the contribution of each individual Xk to the total be small (recall the discussion at the beginning of this section). In fact, since the variable Yn is the sum of n ratios, that is, Sn − E(Sn ) Xk − mk = Vn Vn n

Yn =

k=1

the condition expresses the fact that each individual summand must be uniformly small or, more precisely, that for every ε > 0

|Xk − mk | lim P ≥ε =0 n→∞ Vn

(4.32)

that is, Vn−1 |Xk − mk | → 0[P], which holds whenever eq. (4.31a) holds since 2

ε P(|Xk − mk | ≥ εVn ) = ε

2 Ck

≤

1 Vn2

1 dFk ≤ 2 Vn

n k=1C k

(x − mk )2 dFk Ck

(x − mk )2 dFk → 0

154 Probability theory where the ﬁrst inequality is due to the fact that the domain of integration Ck includes only those x such that |x − mk | ≥ εVn , that is, (x − mk )2 ≥ ε2 Vn2 . Property (4.32) is sometimes called ‘uniform asymptotic negligibility’ (uan). As it should be expected, the iid case of Lindeberg–Levy theorem is just a special case of Proposition 4.23. In fact, if the Xk are iid variables with ﬁnite means m and variances σ 2 , then the sum in (4.31) is simply a sum of n identical terms resulting in 1 σ2

(x − m)2 dF √

{|x−m|≥εσ n}

7 √ 8 which, in turn, must converge to zero because x : |x − mk | ≥ ε n → ∅ as n → ∞. A second special case of Proposition 4.23 occurs when the Xk are uniformly bounded – that is, |Xk | ≤ M for all k – and Vn2 → ∞ as n → ∞. Then

2

ICk (x − mk )2 dFk ≤ (2M)2 P{|x − mk | ≥ εVn }

(x − mk ) dFk = R

Ck

≤

(2M)2 σk2 ε 2 Vn2

where Chebyshev’s inequality (eq. (2.36b)) has been taken into account in the second inequality. From the relations above the Lindeberg condition follows because n 1 (2M)2 (x − mk )2 dFk ≤ 2 2 → 0 2 Vn ε Vn k=1C k

as n → ∞. A third special case of Lindeberg theorem goes under the name of Liapunov’s theorem which can be stated as follows Proposition 4.24(a) (Liapunov’s CLT) Let Xk be a sequence of independent r.v.s with ﬁnite means mk and variances σk2 (k = 1, 2, . . .). If, for some α > 0, 1 Vn2+α

n E |Xk − mk |2+α → 0 k=1

as n → ∞, then Yn → Z[D].

(4.33a)

Limits, convergences and the law of large numbers

155

In fact, Lindeberg’s condition follows owing to the relations 1 1 2 ) dF |x − mk |2+α dFk (x − m ≤ k k 2+α α Vn2 ε Vn k k Ck

≤

Ck

k E(|Xk − mk | ε α Vn2+α

2+α )

where the ﬁrst inequality holds because |x − mk | ≥ εVn . Liapunov’s theorem is sometimes given in a slightly less general form by requiring that ρk = E(|Xk − mk |3 ) < ∞, that is, that all the Xk have ﬁnite third-order central absolute moment. Then Yn → Z[D] if ⎛ ⎞1/3 n 1 ⎝ ⎠ ρk =0 lim n→∞ Vn

(4.33b)

k=1

At this point it is worth noting that eq. (4.31) is a sufﬁcient but not necessary condition for convergence in distribution to Z. This means that there exist sequences of independent r.v.s which converge (weakly) to Z without satisfying Lindeberg’s condition. However, it turns out that for those sequences Xk (of independent r.v.s) such that lim max

n→∞ k≤n

σk2 Vn2

=0

(4.34)

eq. (4.31) is a necessary and sufﬁcient condition for weak convergence to Z. This is expressed in the following proposition Proposition 4.24(b) (Lindeberg–Feller CLT) Let X1 , X2 , . . . be as in Proposition 4.23. Then the Lindeberg condition (4.31) holds if and only if Yn → Z[D] and eq. (4.34) holds. A slightly different version of this theorem replaces eq. (4.34) by the uan condition of eq. (4.32). The interested reader can ﬁnd both the statement and the proof of this theorem in Ref. [1]. 4.6.1

Final remarks

We close this chapter with a few complementary remarks which, although outside our scopes, can be useful to the reader interested in further analysis. The different forms of CLT given above consider D-convergence which, we recall, is a statement on PDFs and, in general, implies nothing on the convergence properties of pmfs or – when they exist – pdfs. However, in

156 Probability theory Section 4.2.1 we mentioned the local deMoivre–Laplace theorem where a sequence of (discrete) Bernoulli pmfs converges to a standardized Gaussian pdf. This is a special case of ‘lattice distributions’ converging to the standardized Gaussian pdf. Without entering into details we only say here that a random variable is said to have a ‘lattice distribution’ if all its values can be expressed in the form a + hk where a, h are two real numbers, h > 0 and k = 0, ±1, ±2,. . .. Bernoulli’s and Poisson’s distributions are just two examples among others. Under the assumption of iid variables with ﬁnite means and variances, a further restriction on h (the requirement of being ‘maximal’) provides a necessary and sufﬁcient condition for the validity of a ‘local’ version of the CLT. The deﬁnition of maximality for h, the theorem itself and its proof can be found in Chapter 8 of [9]. Also, in the same chapter, the following result for continuous variables is given: Proposition 4.25 Let X1 , X2 , . . . be iid variables with ﬁnite means m and variances σ 2 . If, starting from a certain integer n = n0 the variable Yn = √ (σ n)−1 [Sn − n m] has a density fn (x), then 1 fn (x) − √ exp(−x2/2) → 0 2π uniformly for −∞ < x < ∞ if and only if there exists n1 such that fn1 (x) is bounded. A second aspect to consider is whether the Gaussian is the only limiting distribution for sums of independent random variables. The answer to this question is no. In fact, a counterexample has been given in Proposition 4.1 stating that the Poisson distribution is a limiting distribution for binomial r.v.s. Moreover, even in the case of iid variables the requirement of ﬁnite means and variances may not be met. So, in the light of the fact that a so-called Cauchy r.v., whose pdf is f (x) =

1 π(1 + x2 )

(4.35a)

has not a ﬁnite variance, one might ask, for example, if (4.35) could be a limiting distribution or, conversely, what kind of distribution – if any – is the limit of a sequence of independent r.v.s Xk distributed according to (4.35a). Incidentally, we note that (i) the PDF and CF of a Cauchy r.v. are, respectively 1 1 arctan x + π 2 φ(u) = exp(−|u|) F(x) =

(4.35b)

Limits, convergences and the law of large numbers

157

(ii) in (4.35a), failure to converge to the Gaussian distribution is due to the presence of long ‘inverse-power-law’ tails as |x| → ∞. These broad tails, however, do not preclude the existence of a limiting distribution. Limiting problems of the types just mentioned led to the identiﬁcation of the classes of ‘stable’ (or Levy) distributions and of ‘inﬁnitely divisible’ distributions, where the latter class is larger and includes the former. As it should be expected, the Gaussian, the Poisson and the Cauchy distributions are inﬁnitely divisible (the Gaussian and Cauchy distributions, moreover, belong to the class of stable distributions). For the interested reader, more on this topic can be found, for example, in [1, 9, 10, 21]. The third and last remark is on the multi-dimensional CLT for iid random vectors which, in essence, is a straightforward extension of the onedimensional case. In fact, just as the sum of a large number of iid variables is approximately Gaussian under rather wide conditions, similarly the sum of a large number of iid vectors is approximately Gaussian (with the appropriate dimension). In more mathematical terms, we have the following proposition: Proposition 4.26 Let X1 = (X1(1) , . . . , X1(k) ), X2 = (X2(1) , . . . , X2(k) ), . . . be k-dimensional iid random vectors with ﬁnite mean m and covariance matrix K. Denoting by Sn the vector sum

Sn =

n j=1

⎞ ⎛ n n n Xj = ⎝ Xj(1) , Xj(2) , . . . , Xj(k) ⎠ j=1

j=1

(4.36)

j=1

√ then the sequence (Sn − nm)/ n converges weakly to Z, where Z is a k-dimensional Gaussian vector with mean 0 and covariance matrix K.

4.7

Summary and comments

In experiments involving elements of randomness, long-term regularities tend to become clearer and clearer as the number of trials increases and one of the great achievements of probability theory consists in having established the general conditions under which these regularities occur. On mathematical grounds, a tendency towards something implies some kind of limit, although – as is the case in probability – this is not necessarily the familiar limit of elementary calculus. In this light, Sections 4.2 and 4.3 deﬁne a number of different types of convergence by also giving their main individual properties and, when they exist, their mutual relations. Both sections have a subsection – 4.2.1 and 4.3.1, respectively – where additional remarks are made and further details are considered. In essence, the main types of convergences used in probability theory are: convergence in distribution (or weak convergence), convergence in probability, almost-sure convergence and convergence in the kth median

158 Probability theory (k = 1, 2, . . .). Respectively, they are denoted in this text by the symbols D, P, a.s. and Mk convergence and the main mutual relations are as follows: (i) Mk ⇒ Mj (j ≤ k),

(ii) M1 ⇒ P ⇒ D,

(iii) a.s. ⇒ P ⇒ D.

The relation between Mk - and a.s.-convergence is considered in Proposition 4.10 and partial converses of the implications above, when they exist, are also given. With these notions of convergence at our disposal, we then investigate the results classiﬁed under the name LLN, which concern the asymptotic behaviour of the arithmetic mean of a sequence of random variables. In this regard, it is customary to distinguish between weak (WLLN) and strong (SLLN) law of large numbers depending on the type of convergence involved in the mathematical formulation – that is, P- or a.s.–convergence, respectively. The names ‘weak’ and ‘strong’ follow from implication (iii) above and the two laws are considered in Sections 4.4 (WLLN) and 4.5 (SLLN). So, in Section 4.4 we ﬁnd, for instance, Markov’s, Chebychev’s, Khintchine’s and Kolmogorov’s WLLN, the various form differing on the conditions satisﬁed by the sequence of r.v.s involved (e.g. independence, independence and equal probability distributions, ﬁnite variances, etc.). Also, it is shown that Bernoulli’s WLLN – one of the oldest results of probability theory – is a consequence of the more general (and more recent) result due to Markov. It should be noted that most of the above results provide sufﬁcient conditions for the WLLN to hold and only Kolmogorov’s theorem is an ‘if and only if’ statement. Section 4.5 on the SLLN is basically similar to Section 4.4; various forms of SLLN are given and a noteworthy result is expressed by Proposition 4.19 which shows that for iid r.v.s (a frequently encountered case in applications) the existence of a ﬁnite ﬁrst-order moment is a necessary and sufﬁcient condition for the WLLN to hold. In Section 4.5, moreover, we also give two additional results: (a) Borel– Cantelli lemma and (b) Kolmogorov’s inequality. These are two fundamental results of probability theory in general. The main reason why they are included in this section is because they play a key part in the proofs of the theorems on the SLLN, but it must be pointed out that their importance lies well beyond this context. Having established the conditions under which the LLN holds, Section 4.6 turns to one of the most famous results of probability, the so-called CLT which concerns the D-convergence of sequences of (independent) random variables to the normal distribution. We give two forms of this result: Lindeberg–Levy CLT and Lindeberg’s CLT, where this latter result shows that – provided that the contribution of each individual r.v. is ‘small’ – the assumption of identically distributed variables is not necessary. A number of special cases of the theorem are also considered.

Limits, convergences and the law of large numbers

159

The chapter ends with Section 4.6.1 including some additional remarks worthy of mention in their own right. First, some comments are made on ‘local’ convergence theorems, where the term means the convergence of pdfs (or pmfs for discrete r.v.s) to the normal pdf. We recall, in fact, that Dconvergence is a statement on PDFs and does not necessarily imply the convergence of densities. Second, owing to the popularity of the CLT, one might be tempted to think that the normal distribution is the only limiting distribution of sequences of r.v.s. As a matter of fact, it is important to point out that it is not so because there exists a whole class of limiting distributions and the normal is just a member of this class. The subject, however, is outside the scope of the book and references are given for the interested reader. The third and ﬁnal remark is an explicit statement of the multi-dimensional CLT for iid random vectors.

References [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Boccara, N., ‘Probabilités’, Ellipses – Collection Mathématiques Pour l’Ingénieur (1995). [3] Brémaud, P., ‘An Introduction to Probabilistic Modeling’, Springer-Verlag, New York (1988). [4] Breiman, L., ‘Probability’, SIAM – Society for Industrial and Applied Mathematics, Philadelphia (1992). [5] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press,19th printing (1999). [6] Dall’Aglio, G., ‘Calcolo delle Probabilità’, Zanichelli, Bologna (2000). [7] Duncan, A.J., ‘Quality Control and Industrial Statistics’, 5th edn., Irwin, Homewood, Illinois. [8] Friedman, A., ‘Foundations of Modern Analysis’, Dover Publications, New York (1982). [9] Gnedenko, B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [10] Gnedenko, B.V., Kolmogorov, A.N., ‘Limit Distributions for Sums of Independent Random Variables’, Addison-Wesley, Reading MA (1954). [11] Heathcote, C.R., ‘Probability: Elements of the Mathematical Theory’, Dover Publications, NewYork (2000). [12] Jacod J., Protter, P., ‘Probability Essentials’, 2nd edn., Springer-Verlag, Berlin (2003). [13] Karr, R.A., ‘Probability’, Springer Texts in Statistics, Springer-Verlag, New York (1993). [14] Klimov, G., ‘Probability Theory and Mathematical Statistics’, Mir Publishers, Moscow (1986). [15] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [16] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000).

160 Probability theory [17] Pfeiffer, P.E., ‘Concepts of Probability Theory’, 2nd edn., Dover Publications, New York (1978). [18] Biswas, S., ‘Topics in Statistical Methodology’, Wiley Eastern Limited, New Delhi (1991). [19] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997). [20] Ventsel, E.S., ‘Teoria delle Probabilità’, Mir Publisher, Moscow (1983). [21] Wolfgang, P., Baschnagel, J., ‘Stochastic Processes, from Physics to Finance’, Springer-Verlag, Berlin (1999).

Part II

Mathematical statistics

5

5.1

Statistics: preliminary ideas and basic notions

Introduction

With little doubt, the theory of probability considered in the previous chapters is an elegant and consistent mathematical construction worthy of study in its own right. However, since it all started out from the need to obtain answers and/or make predictions on a number of practical problems, it is reasonable to expect that the abstract objects and propositions of the theory must either have their counterparts in the physical world or express relations between real-world entities. As far as our present knowledge goes, real-world phenomena are tested by observation and experiment and these activities, in turn, produce a set – or sets – of data. With the hope to understand the phenomena under investigation – or at least of some of their main features – we ‘manipulate’ these data in order to extract the useful information. In experiments where elements of randomness play a part, the manipulation process is the realm of ‘Statistics’ which, therefore, is a discipline closely related to probability theory although, in solving speciﬁc problems, it uses techniques and methods of its own. Broadly speaking, the main purposes of statistics are classiﬁed under three headings: description, analysis and prediction. In most cases, clearly, the distinction is not sharp and these classes are introduced mainly as a matter of convenience. The point is that, in general, the individual data are not important in themselves but they are considered as a means to an end: the measure of a certain physical property of interest, the test of a hypothesis or the prediction of future occurrences under given conditions. Whatever the ﬁnal objectives of the experiment, statistical methods are techniques of ‘inductive inference’ in which a particular set (or sets) of data – the so-called ‘realization of the sample’ – is used to draw inferences of general nature on a ‘population’ under study. This process is intrinsically different and must be distinguished from ‘deductive inference’ where conclusions based on partial information are always correct, provided that the original information is correct. For example, in basic geometry the examination of particular cases leads to the deduction that the sum of the angles

164 Mathematical statistics of a triangle equals 180 degrees, a conclusion which is always correct within the framework of plane geometry. By contrast, inductive inferences drawn from incomplete information may be wrong even if the original information is not. In the ﬁeld of statistics, this possibility is often related to the (frequently overlooked) process of data collection on one hand – it is evident that insufﬁcient or biased data and/or the failure to consider an important inﬂuencing factor in the experiment may lead to incorrect conclusions – and, on the other hand, to the fact that in general we can only make probabilistic statements and/or predictions. By their own nature, in fact, statements or predictions of this kind always leave a ‘margin of error’ even if the data have been properly collected. To face this problem, it is necessary to design the experiment in such a way as to reduce this uncertainty to values which may be considered acceptable for the situation at hand. Statistics itself, of course, provides methods and guidelines to accomplish this task but the analyst’s insight of the problem is, in this regard, often invaluable. Last but not least, it should always be kept in mind that an essential part of any statistical analysis consists in a quantitative statement on the ‘goodness’ of our inferences, conclusions and/or results. A ﬁnal remark is not out of place in these introductory notes. It is a word of caution taken from Mandel’s excellent book [20]. In the light of the fact that statistical results are often stated in mathematical language, Mandel observes that ‘the mathematical mode of expression has both advantages and disadvantages. Among its virtues are a large degree of objectivity, precision and clarity. Its greatest disadvantage lies in its ability to hide some very inadequate experimentation behind a brilliant facade’. In this regard, it is surely worth having a look at Huff’s fully enjoyable booklet [15].

5.2

The statistical model and some notes on sampling

As explained in Chapter 4, the mathematical models of probability are based on the notion of probability space (W, S, P), where W is a non-empty set, S a σ -algebra of subsets of W (the ‘events’) and P is a probability function deﬁned on S. Moreover, an important point is that one generally considers – more or less implicitly – P to be fully deﬁned. In practice, however, P is seldom known fully and there exists some degree of uncertainty attached to it. Depending on the problem, the degree of uncertainty may vary from a situation of complete indeterminacy – where P could be any probability function that can be deﬁned on S – to cases of partial indeterminacy in which P is known to belong to a given class but we lack some information which, were it available, would specify P completely. In general terms, the goal of statistics is to reduce the uncertainty in order to gain information and/or make predictions on the phenomena under investigation. This task, as observed in the introduction, is accomplished by using and ‘manipulating’ the data collected in experiment(s).

Preliminary ideas and notions

165

In more mathematical terms, the general idea is as follows. We perform an experiment consisting of n trials by assuming that the result xi (i = 1, 2, . . . , n) of the ith trial is associated to a random variable Xi . By so doing, we obtain a set of n observations (x1 , x2 , . . . , xn ) – the so-called realization of the sample – associated to the set of r.v.s (X1 , X2 , . . . , Xn ) which, in turn, is called a random sample (of size n). Both quantities can be considered as n-dimensional vectors and denoted by x and X, respectively. Also, we call sample space the set of all possible values of X and this, depending on X, may be the whole Rn or part of it (if X is continuous) or may consist in a ﬁnite or countable number of points of Rn (if X is discrete). With the generally implicit assumption that there exists a collection of subsets of forming a σ -algebra – which is always the case in practice – one deﬁnes (, ) as the statistical model of the experiment, where here denotes the class of possible candidates of probability functions pertaining to the sample X. Clearly, one of the elements of will be the (totally or partially unknown) ‘true’ probability function PX . Now, referring back for a moment to Chapters 2 and 3, we recall that a random vector X on (W, S, P) deﬁnes a PDF FX which, in turn, completely determines both the ‘induced’ probability PX and the original probability P. Any degree of uncertainty on P, therefore, will be reﬂected on FX (or, as appropriate, on the pmf pX or pdf fX , when they exist) so that, equivalently, we can say that our statistical model is deﬁned by (, ), where is a class of PDFs such that FX (x) = PX (X1 ≤ x1 , . . . , Xn ≤ xn ) ∈ . A particular but rather common situation occurs when the experiment consists in n independent repetitions of the same trial (e.g. tossing a coin, rolling a die, measuring n times a physical quantity under similar conditions, etc.). In this case the components of the sample X1 , X2 , . . . , Xn are iid random variables, that is, they are mutually independent and are all distributed like some r.v. X so that FXi (xi ) = FX (xi ) for all i = 1, 2, . . . , n. The variable X is often called the parent random variable and the set RX of all possible values of X is called the population; also, with this terminology, one can call X a ‘sample (of size n) from the population RX ’. So, depending on the problem at hand, there are two possibilities (1) the PDF FX is totally unknown (and therefore, a priori, may include any PDF), or (2) the general type of FX is known – or assumed to be known – but we lack information on a certain parameter θ whose ‘true’ value may vary within a certain set (note that, in general, θ may be a scalar – then ⊆ R – or a k-dimensional vector (k = 2, 3, . . .), and then ⊆ Rk ). In case (1) our interest may be (1a) to draw inferences on the type of PDF underlying the phenomena under study or (1b) to draw inferences which do not depend on the speciﬁc distribution of the population from which

166 Mathematical statistics the sample is taken. Statistical techniques that are – totally or partially – insensitive to the type of distribution and can be applied ignoring this aspect are called non-parametric or distribution-free methods. In case (2) we speak of parametric model and the class is of the form = {F(x; θ ) : θ ∈ }

(5.1)

where, for every ﬁxed θ ∈ , F(x; θ ) is a well-deﬁned PDF (the semicolon between x and θ denotes that F is a function of x with θ as a parameter, and not a function of two variables). Clearly, one denotes by Pθ the probability function associated to F(x; θ ). In most applications, parametric models are either discrete of absolutely continuous, depending on the type of PDFs in . It is evident that in these cases the model can be speciﬁed by means of a class of pmfs or pdfs, respectively. Example 5.1 (Parametric models) Two examples may be of help to clarify the theoretical discussion above. (i) Consider the experiment of tossing n times a coin whose bias, if any, is unknown. Assuming that we call a head a ‘success’, the natural parent r.v. X associated with the experiment assigns the value 1 to a success and 0 to a failure (tail). Since the experiment is a Bernoulli scheme, the distribution of X will clearly be a binomial pmf (eq. (2.41a)) whose probability of success, however, is not known. Then, our statistical model consists of two sets: , which includes any possible sequence (of n elements) of 1s and 0s, and which includes all the pmfs of the type n x p(x; θ ) = θ (1 − θ )n−x x

(5.2)

where x denotes the number of successes (x = 0, 1, . . . , n), and θ ∈ = [0, 1] because the probability of success can be any number between 0 and 1 (0 and 1 representing the case of totally biased coin). Performing the experiment once leads to a realization of the sample – that is, one element of – which form our experimental data. Statistics, using these data, provides methods of evaluating – estimating is the correct term – the unknown parameter θ, that is, to make inferences on how much biased is the coin. (ii) From previous information it is known that the length of the daily output – say, 5000 pieces – of a machine designed to cut metal rods in pieces of nominal length = 1.00 m, follows a Gaussian distribution. The mean and variance of the distribution, however, are unknown. In this case X is the length of a rod and consists of the density functions f (x; θ1 , θ2 ) =

−1 % √ $ 2π θ2 exp − (x − θ1 )2 /2θ22

(5.3)

Preliminary ideas and notions

167

where, in principle, θ1 ∈ 1 = (−∞, ∞) and θ2 ∈ 2 = (−∞, ∞) but of course – being θ1 , θ2 the mean and standard deviation of the process – the choice can in reality be restricted to much smaller intervals of possible values. By selecting n, say n = 50, pieces of a daily production and by accurately measuring their lengths we can estimate the two parameters, thus drawing inferences on the population (the lengths of the 5000 daily pieces). The realization of the sample are our experimental data, that is, the 50 numbers (x1 , x2 , . . . , x50 ) resulting from the measurement. Although the type of parametrization is often suggested by the problem, it should be noted that it is not unique. In fact, if h : → is a one-to-one function, the model (5.1) can be equivalently written as = {F(x; ψ) : ψ ∈ }

(5.4)

where = {ψ : ψ = h(θ ), θ ∈ } and the choice between (5.1) and (5.2) is generally a matter of convenience. One word of caution, however, is in order: some inferences are not invariant under a change of parametrization, meaning in other words that there are statistical techniques which are affected by the choice of parametrization. This point will be considered in due time if and whenever needed in the course of future discussions. It is worth at this point to pay some attention to the process by which we collect our data, that is, the so-called procedure of sampling. Its importance lies in the fact that inappropriate sampling may lead to wrong conclusions because our inferences cannot be any better than the data from which they originate. If the desired information is not implicitly contained in the data, it will never come out – no matter how sophisticated the statistical technique we adopt. Moreover, if needed, a good set of data can be analysed more that once by using different techniques, while a poor set of data is either hopeless or leads to conclusions which are too vague to be of any practical use. At the planning stage, therefore – after the goal of the experiment has been clearly stated – there are a certain number of questions that need an answer. Two of these, the most intuitive, are: ‘how do we select the sample?’ and ‘of what size?’. In regard to the ﬁrst question the basic prerequisite is that the sample must be drawn at random. A strict deﬁnition of what exactly constitutes a ‘random sample’ is rather difﬁcult to give but, luckily, it is often easier to spot signs of the contrary and decide that a given procedure should be discarded because of non-randomness. The main idea, clearly, is to avoid any source of bias and make our sample, as it is often heard, ‘representative’ of the population under study. In other words, we must adopt a sampling method that will give every element of the population an equal chance of being drawn. In this light, the two simplest sampling schemes are called ‘simple sampling (with replacement)’ and ‘sampling without replacement’. In both cases the sampling procedure is very much like drawing random tickets from an urn: in

168 Mathematical statistics the ﬁrst case we draw a ticket, note the value inscribed, and replace the ticket in the urn while in the second case we do not replace the ticket before the next drawing. It is worth noting that the main difference is that sampling without replacement, in general, cannot be considered as a repetition of a random experiment under uniform conditions because the composition of the population changes from one drawing to another. However, if (i) the population is inﬁnite or (ii) very large and/or our sample consists in a small fraction of it (as a rule of thumb, at most 5%), the two schemes are essentially the same because removal of a few items does not signiﬁcantly alter – case (ii) – the composition of the population or – case (i) – does not alter it at all. The two sampling schemes mentioned above are widely used although, clearly, they are not the only ones and sometimes more elaborate techniques are used in speciﬁc applications. For our purposes, we will generally assume the case of simple sampling, unless otherwise explicitly stated in the course of future discussions. In regard to random samples, a ﬁnal point worthy of mention is that, for ﬁnite populations, the use of (widely available) tables of random numbers is very common among statisticians. The members of the population are associated with the set of random numbers or some subset thereof; then a sample is taken from this set – for example, by blindly putting a pencil down on the table and picking n numbers in that section of the table – and the corresponding items of the population are selected. In case of sampling without replacement we must disregard any number that has already appeared. The number n is the size of the sample which, as noted above, is one of the main points to consider at the planning stage because its value – directly or indirectly – affects the quality and accuracy of our conclusions. However, since the role of the sample size will become clearer as we proceed, further considerations will be made in due time.

5.3

Sample characteristics

It often happens that an experiment consists in performing n independent repetitions of a trial to which a one-dimensional parent r.v. X, with PDF FX , is attached. Then, the sample is the sequence of iid r.v.s X1 , . . . , Xn and the realization of the sample will be is a sequence x1 , x2 , . . . , xn of n observed values of X. Recalling from Section 2.3.2 the numerical descriptors of a r.v. – that is, mean, variance, moments, central moments, etc. – we can deﬁne the sample, or statistical, counterparts of these quantities. So, the ordinary (i.e. non-central) sample moment and sample central moment of order k (k = 1, 2, . . .) are 1 k Xi n

(5.5)

1 (Xi − A1 )k n

(5.6)

n

Ak =

i=1 n

Ck =

i=1

Preliminary ideas and notions

169

respectively, where A1 is the sample mean and C2 is the sample variance. The speciﬁc values of these quantities obtained from a realization of the sample x1 , x2 , . . . , xn will be denoted by the corresponding lowercase letters, that is, ak and ck , respectively. So, for instance, if we repeat the experiment (i.e. other n trials) a second time – thus obtaining a new set x1 , x2 , . . . , xn of observed values – we will have, in general ak = ak and ck = ck (k = 1, 2, . . .). At this point, in order not to get lost in symbols, a few comments on notation are necessary: (a) the sample characteristics are denoted by Ak and Ck to distinguish them from their population (or theoretical) counterparts E(X k ) – with the mean E(X) = µ as a special case – and E[(X − µ)k ] – with the variance E[(X−µ)2 ] = σ 2 (or Var(X)) as a special case. The parent r.v. X and the sample size n to which Ak and Ck refer are often clear from the context and therefore will be generally omitted unless necessary either to avoid confusion or to make a point. (b) Since some population characteristics are given special Greek symbols – for example, µ, σ 2 and the standard deviation σ – it is customary to indicate their sample counterparts by the corresponding uppercase italic letters, that is, M, S2 and S, respectively. So, in the light of eqs (5.5) and (5.6) we have S2 = C2 = n−1

M = A1 ;

, (Xi − M)2 and, clearly, S = S2 . i

(c) Italic lowercase letters, m, s2 and s, denote the speciﬁc realization of the sample characteristic obtained as a result of the experiment, that is: m = n−1

i

xi ; s2 = c2 = n−1

(xi − m)2 and s =

, s2

i

(d) Greek letters will be often used for higher-order population characteristics. So, αk and µk will denote respectively the (population) ordinary and central moments of order k. In this light, clearly, α1 = µ and µ2 = σ 2 but for these lower-order moments the notation µ and σ 2 (or Var(X)) will generally be preferred. The main difference to be borne in mind is that the population characteristics are ﬁxed (though sometimes unknown) constants while the sample characteristics are conceived as random variables whose realizations are obtained by actually performing the experiment. More generally, since Ak and Ck are just special cases of (measurable) functions of X1 , . . . , Xn , the above considerations apply to any (measurable) function G(X1 , . . . , Xn ) of the sample. Any function of this type which contains no unknown parameters is often called a statistic. So, for instance, the Ck deﬁned in (5.6) are statistics while the quantities n−1 i (Xi − µ)k are not if µ is unknown.

170 Mathematical statistics Returning to the main discussion, a ﬁrst observation to be made is that the relations between theoretical moments given in previous chapters still hold true for their sample counterparts and for their realizations as well. Then, for example, by appropriately changing the symbols, eq. (2.34) becomes S2 = A2 − A21 = A2 − M2

(5.7)

or, more generally, the relation (2.33) between central and ordinary moments is Ck =

k (−1)j k! j M Ak−j j!(k − j)!

(5.8)

j=0

and similar equations hold between ak and ck . Moreover, conceiving the sample characteristics as random variables implies that they will have a probability distribution in their own right which, as should be expected, will be determined by FX . Whatever these distributions may be, the consequence is that it makes sense to speak, for instance, of the mean, variance and, in general, of moments of the sampling moments. Let us start by considering the mean E(M) of the sample mean M. Since E(Xi ) = µ for all i = 1, . . . , n, the properties of expectation give 1 1 E(M) = E Xi = E(Xi ) = µ n n i

(5.9)

i

The variance of M, in turn, can be obtained by using eq. (3.116) and the independence of the Xi . Therefore µ2 (M) = Var(M) =

1 1 σ2 Var(X ) = n Var(X) = i n n2 n2

(5.10)

i

√ and the standard deviation is σM = σ/ n. Similarly, it is left to the reader to show that the third- and fourth-order central moments of M are given by % µ $ 3 µ3 (M) ≡ E (M − µ)3 = 2 n % µ $ 3(n − 1) 4 4 µ4 (M) ≡ E (M − µ)4 = 3 + σ n n3

(5.11a)

and so on, with more tedious calculations, for the ﬁfth, sixth order, etc. It is useful, however, to know the order of magnitude of the leading term in

Preliminary ideas and notions

171

these central moments of M; for even and odd moments we have µ2m (M) = O(n−m ), µ2m−1 (M) = O(n−m ),

m = 1, 2, . . .

(5.11b)

respectively, where the symbol O(q) is well known from analysis and means ‘of the same order of magnitude’ of the quantity q in parenthesis. Equation (5.11b) can be checked by looking at (5.10) and (5.11a). Next, turning variance S2 , eq. (5.7) gives our2 attention2 to the sample 2 −1 2 2 E(S ) = n E i Xi − E(M ) = E(X ) − E(M ). Then, using eq. (2.34) for both r.v.s X and M we get its mean as

2 E(S2 ) = σ 2 + µ2 − σM − µ2 = σ 2 −

n−1 2 σ2 = σ n n

(5.12)

a bit more involved. DeﬁningYi = The calculation of the variance of S2 is ¯ 2 , where Y¯ = n−1 Xi − µ(i = 1, 2, . . . , n) we get S2 = n−1 i (Yi − Y) j Yj . Then we can write ⎡ 2 ⎤ 1 1 1 ¯ 2= ⎣ S2 = (Yi − Y) Yi2 − Yi ⎦ n n n i i i ⎤ ⎡ 1 1 2 2 = ⎣ Yi2 − Yi − Yi Yj⎦ n n n i

=

i

n−1 n2

i

i<j

2 Yi2 − 2 Yi Yj n i<j

Squaring this quantity and taking its expectation gives E[(S2 )2 ] =

2

n−1 n2 ⎡⎛

⎡ 2 ⎤ E⎣ Yi2 ⎦ i

⎞2 ⎤ ⎤ ⎡ 4 ⎢⎝ 4(n − 1) ⎥ + 4E⎣ Yi Y j ⎠ ⎦ − E⎣ Yr2 Yi Yj ⎦ n n4 r i<j

i<j

172 Mathematical statistics Now, taking independence into account plus the fact that E(Yi ) = 0 for all i, the last term on the r.h.s. is zero while the ﬁrst and second term lead to ⎞ ⎛

n − 1 2 ⎝ 4 Yi + 2 Yi2 Yj2 ⎠ E n4 i⎞ i<j ⎛ 4 ⎝ Yi2 Yj2 ⎠ E n4

i<j

respectively. Therefore ⎞ 2 2+4 2(n − 1) (n − 1) E[(S2 )2 ] = E ⎝ Yi4 + Yi2 Yj2 ⎠ n4 n4 ⎛

i

=

(n − 1)2 n3

µ4 +

i<j

(n − 1)2

+2

n3

(5.13)

(n − 1)σ 4

where the last equality holds because E(Yi4 ) = µ4 , E(Yi2 ) = σ 2 and there are n(n − 1)/2 combinations of n r.v.s taken two at a time. Finally, since Var(S2 ) = E[(S2 )2 ] − E2 (S2 ), we use eqs (5.13) and (5.12) to get µ2 (S2 ) = Var(S2 ) =

(n − 1)2 n3

µ4 −

n−3 4 σ n−1

(5.14a)

So, in particular, if the original population is normal then the mean and variance of S2 are given, respectively, by eq. (5.12) and by Var(S2 ) =

2(n − 1) 4 σ n2

(5.14b)

where this last result follows from (5.14a) by taking eq. (2.42d) into account. With rather cumbersome calculations one could then go on to obtain µ3 (S2 ), µ4 (S2 ), etc. We do not do it here but limit ourselves to two further results worthy of mention: the ﬁrst concerns the mean and variance of the sample moments Ak and their covariances. It is rather easy to determine E(Ak ) = αk

% α − α2 $ Var(Ak ) ≡ E (Ak − αk )2 = 2k n k $ % α −α α Cov(Ak Al ) ≡ E (Ak − αk )(Al − αl ) = k+l n k l

(5.15)

where the ﬁrst two equations are in agreement with the special cases (5.9) and (5.10) when one notes that A1 = M, α1 = µ and α2 = σ 2 + µ2 . For the

Preliminary ideas and notions

173

order of magnitude of even and odd central moments of the Ak we have % $ µ2m (Ak ) ≡ E (Ak − ak )2m = O(n−m ), % $ µ2m−1 (Ak ) ≡ E (Ak − ak )2m−1 = O(n−m ),

m = 1, 2, . . .

(5.16)

and it is easily seen that eq. (5.11b) are the special case k = 1 of (5.16). The second result gives the covariance between the sample mean and the sample variance; it is left to the reader to show that *

+ n−1 2 n−1 E (M − µ) S2 − σ µ3 = E[(M − µ)S2 ] = n n2

(5.17)

which implies that for any symmetric distribution M and S2 are uncorrelated. In fact, as it is probably known to the reader, µ3 is a measure of skewness – or asymmetry or lopsidedness – of the distribution so that µ3 = 0 for any symmetric distribution. More speciﬁcally, the (adimensional) coefﬁcient of skewness γ1 is often used, where by deﬁnition γ1 =

µ3 3/2 µ2

=

µ3 σ3

(5.18)

With the above results at hand, one can determine the mean and variance of a number of (well-behaved) functions of sample moments by using the approximations given in Section 3.5.1. So, for example, if k, l ≥ 1 are any two integers and g(Ak , Al ) is a twice differentiable function in some neighbourhood of (αk , αl ), then eq. (3.126) gives E[g(Ak , Al ) ∼ = g(αk , αl )

(5.19a)

while eq. (3.128a) leads to Var[g(Ak , Al )] ∼ =

∂g ∂Ak

+2

2

Var(Ak ) +

∂g ∂Al

∂g ∂g Cov(Ak Al ) ∂Ak ∂Al

2 Var(Al ) (5.19b)

where it is understood that all derivatives are calculated at the point (αk , αl ). Note, in particular, that eqs (5.19a) and (5.19b) can be used to approximate the mean and variance of the sample central moment Ck which, as shown by eq. (5.8), is a polynomial in Ak , Ak−1 , . . . , A1 . Example 5.2(a) Consider the mean and variance of C2 = A2 − A21 . From eq. (5.19a) we get E(C2 ) = α2 − µ2 = σ 2 , which is the leading term in the

174 Mathematical statistics exact result (5.12). On the other hand, eq. (5.19b) yields Var(C2 ) ∼ =

µ4 − µ22 α4 − α22 + 8µ2 α2 − 4µ4 − 4µα3 = n n

which, as it should be expected, is the leading term of eq. (5.14) (the second equality is obtained by taking into account the relations between ordinary and central moments). It is evident that the method leading to eqs (5.19a) and (5.19b) is essentially of analytical nature and, as such, it applies to all cases in which the assumptions of the relevant theorems are satisﬁed. These assumptions, in general, have do with the behaviour of the function g and, for this point, the reader is referred to books of mathematical analysis. Example 5.2(b) As a second example, we calculate√the variance of the socalled (sample) ‘coefﬁcient of variation’ V = S/M = C2 /A1 , provided that this quantity is bounded. We have 1 ∂V = √ ∂C2 2µ µ2 √ µ2 ∂V =− 2 ∂A1 µ so that using eq. (5.19b) and retaining only the leading terms in Var(C2 ), Var(A1 ) and Cov(C2 A1 ) – see eqs (5.14), (5.10) and (5.17), respectively – we get Var(V) ∼ =

µ2 (µ4 − µ22 ) − 4µ3 µ2 µ + 4µ32 4nµ2 µ4

(5.20)

By similar calculations one could obtain, for instance, the approximate mean and variance of the sample counterpart of the coefﬁcient of skewness (5.18). 5.3.1

Asymptotic behaviour of sample characteristics

The considerations of the preceding section readily extend to the multidimensional case and the reader is invited to work out the details. Here we turn our attention to another issue: the asymptotic behaviour of sample characteristics as n tends to inﬁnity or, in practical applications, for large samples. Starting with the sample mean M, we can use Markov’s WLLN (Proposition 4.13) to determine that M → E(M)[P]. In fact, since the Xi are iid r.v.s and Sn = X1 + · · · + Xn = nM, then Var(Sn ) = n2 Var(M) = nσ 2

Preliminary ideas and notions

175

)/n2

and Var(Sn → 0 as n → ∞, showing that the assumptions of the theorem are all satisﬁed. Then, by virtue of eq. (5.9) we have M → µ[P]. Actually, by recalling the various form of SLLN given in Section 4.5, one can state the stronger result M → µ[a.s.]. More generally, as a consequence of Chebyshev’s inequality (see also Proposition 4.12) and the ﬁrst of eq. (5.15) we have Ak (n) → E(Ak ) = αk [P] and, even more, by virtue of Proposition 4.19 Ak (n) → αk [a.s.] whenever αk is ﬁnite. Note that here the n in parenthesis stresses the fact that the moments Ak depend on the sample size. Clearly, similar statements are valid for the sample central moments and for any sample characteristic which is a continuous function of a ﬁnite number of the Ak . These convergence properties, in turn, imply that for large values of n the quantities calculated using the data of the experiment can be regarded as ‘estimates’ of the corresponding population characteristics. However, according to certain criteria used to evaluate the quality of the approximation, we will see in later sections that these may not always be the ‘best’ estimates one can ﬁnd. A second aspect to consider is the fact that the quantity nAk = i Xik is a sum of n independent variables – the Xik – which are independent by virtue of Proposition 3.3 and all have the same distribution. As a consequence, it follows that Proposition 5.1(a)

As n → ∞, the standardized variable

√ nA − nαk n(A − αk ) = ) k ) k n(α2k − αk2 ) α2k − αk2 tends in distribution to the standard Gaussian r.v. In fact, since eq. (5.15) imply E(Xik ) = αk and Var(Xik ) = α2k − αk2 for all i = 1, . . . , n, the result follows from Lindeberg–Levy CLT (Proposition 4.22). Also, note that Proposition 5.1 can be stated in different words by saying that Ak is asymptotically normal with mean αk and variance (α2k − αk2 )/n so that, in particular, the sample mean M is asymptotically normal with mean µ and variance σ 2 /n (see also remark (c) after the proof of Proposition 4.22). In this regard, moreover, when sampling from a normal population we have the special result: Proposition 5.1(b) If the parent r.v. X is normal with mean µ and variance σ 2 , M is exactly normal with mean µ and variance σ 2 /n.

176 Mathematical statistics The proof is immediate if we turn to CFs and note that ϕM (u) = E{exp[iu(X1 + · · · + Xn )/n]} = E[iuX1 /n] · · · E[iuXn /n] = [ϕX (u/n)]n Then, since ϕX has the form given in (2.52), ϕM is the CF of a Gaussian r.v. with mean µ and variance σ 2 /n. Continuing along the above line of reasoning, we can use Proposition 4.26 (the multi-dimensional CLT) to show that Proposition 5.2 The joint distribution of any ﬁnite number of sample moments is itself asymptotically normal. In fact, considering the two-dimensional case for simplicity, let r, s ≥ 1 be any two integers; the vector n(Ar , As )T can be written as r r r X Ar X1 Xn n = i is = + · · · + s As X Xns X i i 1 where Xi = (Xir , Xis )T are n iid two-dimensional vectors such that for all i = 1, . . . , n we have the mean E(Xi ) = (αr , αs )T and the covariance matrix

Var(Xir ) K= Cov(Xis Xir )

Cov(Xir Xis ) α2r − αr2 = s Var(Xi ) αr+s − αr αs

αr+s − αr αs α2s − αs2

where the proof of the relation Cov(Xir Xis ) = αr+s −αr α√ s is immediate. Then, Proposition 4.26 states that, as n → ∞, the vector n(Ar − αr , As − αs ) tends in distribution to a Gaussian two-dimensional vector with mean 0 and covariance matrix K. The extension to a higher dimensional case is straightforward. Another important result is as follows: Proposition 5.3 Let g(x, y) be a twice differentiable function in some neigh√ bourhood of (αr , αs ). Then, as n → ∞, the r.v. n[g(Ar , As )−g(αr , αs )] tends in distribution to a normal variable with zero mean and variance DT KD, where D is the vector

∂g/∂Ar D= ∂g/∂As and it is understood that all derivatives are calculated at the point (αr , αs ). In order to sketch the proof, set cr = ∂g/∂Ar and cs = ∂g/∂As . Since g(Ar , As ) − g(αr , αs ) = cr (Ar − αr ) + cs (As − αs ) + · · ·

Preliminary ideas and notions 177 √ the variable n[g(Ar , As )−g(αr , αs )] is a sum of two r.v.s which, by virtue of Proposition 5.2, tend in distribution to a normal r.v. with zero mean, variances cr2 (α2r − αr ) and cs2 (α2s − αs ) and covariance cr cs (αr+s − αr αs ). Then, Proposition 5.3 follows from the fact that the sum of two dependent normal r.v.s A, B with means a, b and variances σA , σB is itself normal with mean a + b and variance σA2 + σB2 + 2Cov(A, B) (see eq. (3.60)). Also note that one can equivalently state the theorem by saying that g(Ar , As ) is asymptotically normal with mean g(αr , αs ) and variance n−1 DT KD where n−1 K is the covariance matrix of the sample moments Ar , As . The extension to cases where g is a function of more than two moments is immediate and, by appropriately deﬁning D, the matrix notation DT KD still applies. Example 5.3 The sample central moments Ck are functions of A1 , A2 , . . . , Ak and therefore Proposition 5.3 includes them as special cases. In fact, any Ck is asymptotically normal with mean µk and variance 1 µ2k − 2kµk−1 µk+1 − µ2k + k2 µ2 µ2k−1 n

(5.21)

where eq. (5.21) can be obtained starting from eq. (5.8) and noting that the central moments do not depend on where we take the origin. Therefore, there is no loss of generality in assuming the origin at the population mean – that is, setting µ = 0 – so that αk = µk and all derivatives ∂Ck /∂Aj are zero except ∂Ck /∂Ak = 1 and ∂Ck /∂A1 = −kµk−1 . Then, since n−1 DT KD = Var(Ak ) + k2 µ2k−1 Var(A1 ) − 2kµk−1 Cov(Ak A1 ) the desired result follows by taking eq. (5.15) into account. So, for instance, the asymptotic variance of C2 is n−1 (µ4 − µ22 ) which, as expected, coincides with the leading term of eq. (5.14a). Returning to our main discussion, it is worth pointing out that the considerations above do not imply that asymptotic normality – although rather common – is a general rule. In order to give an example of sample characteristics which show a different behaviour in the limit of n → ∞ , we must ﬁrst introduce the notion of ‘order statistics’. Suppose, for simplicity, that we are sampling from a continuous population; each realization of the sample x1 , . . . , xn can be arranged in increasing order x(1) ≤ x(2) ≤ · · · ≤ x(n) where, clearly, x(1) = min(x1 , . . . , xn ) and x(n) = max(x1 , . . . , xn ). Then, letting X(k) , k = 1, . . . , n, denote the r.v. that has the value x(k) for each realization of the sample, we deﬁne a new sequence X(1) , X(2) , . . . , X(n) of random variables satisfying X(1) ≤ · · · ≤ X(n) . This new sequence is called the ordered series of the sample and X(k) , in turn, is called the kth order statistic where, in particular, X(1) and X(n) are the extreme values of the sample. A ﬁrst observation is that order statistics are

178 Mathematical statistics not independent because information on one r.v. of the series provides information on other r.v.s: in fact, for example, if X(k) ≥ x then we know that X(k+1) , . . . , X(n) ≥ x. A second observation is that sampling from an absolutely continuous populations prevents the possibility of two or more order statistics being equal since the probability of, say, X(k) = X(k+1) is zero, thus justifying the expression ‘for simplicity’ at the beginning of this paragraph. The PDF of X(k) can be obtained by noting that the event X(k) ≤ x occurs whenever at least k out of the n independent r.v.s X1 , . . . , Xn are ≤ x. Each one of these events has probability F(x) – where F(x) is the PDF of the parent r.v. X and f (x) = F (x) is its pdf. Therefore we have a binomial PDF given by F(k) (x) =

n n j=k

j

[F(x)]j [1 − F(x)]n−j

(5.22a)

which is absolutely continuous if F(x) is. Taking the derivative with respect to x leads to the pdf of X(k) , that is, F(k)

=

n n j=1

j

jF

j−1

(1 − F)

n−j

n n f− (n − j + 1)F j (1 − F)n−j−1 f j−1 j=k+1

+ n * n n n j− (n − j + 1) = kF k−1 (1 − F)n−k f + j j−1 k j=k+1

× F j−1 (1 − F)n−j f and since the term within square brackets is zero we get n f(k) (x) = k[F(x)]k−1 [1 − F(x)]n−k f (x) k

(5.22b)

where it should be noted that in the extreme cases k = 1 and k = n, respectively, eq. (5.22a) reduces to eqs (2.73a) and (2.72a) while (5.22b) agrees with eqs (2.73b) and (2.72b). In order to investigate the behaviour of order statistics as n → ∞ we must distinguish between mid-terms and extremal terms of the ordered series. We call mid-terms the elements whose index is of the form k = [pn] where p is any ﬁxed number 0 < p < 1 and the notation [a] indicates the integer value of the number a. So, in these cases k depends on n and k/n → p as n → ∞. On the other hand, we call extremal terms the elements of the series whose ordinal index is considered ﬁxed throughout the limiting process and has either the form k = r or k = n − s + 1, where r, s are any two ﬁxed integers ≥1. Note that k = n − s + 1 always indicates the sth element from the top, irrespective of the sample size n which, in fact, is assumed to increase indeﬁnitely.

Preliminary ideas and notions

179

Without entering into the details of the calculation, it can be shown (see, for example, [3] or [19]) that the mid-terms are asymptotically normal. However, the extremal terms are not. In fact, for instance, consider X(r) and deﬁne the new variable γ = nF(x), denoting by g(r) (γ ) its pdf. Since we must have f(r) (x) dx = g(r) (γ ) dγ , we get from (5.22b) g(r) (γ ) =

γ n−r γ n−r r n γ r−1 r n γ r 1− 1− = n r n n γ r n n (5.23)

and – being 0 ≤ γ /n ≤ 1 – we can use the limit (4.9) to obtain lim g(r) (γ ) =

n→∞

γ r−1 −γ γ r−1 −γ e = e (r − 1)! (r)

(5.24)

where in the second expression we used the well-known ‘gamma function’ deﬁned in Appendix C. Similarly, if for the sth statistic from the top X(n−s+1) one deﬁnes g = n(1 − F(x)) it is easy to determine that g(s) (γ ) is again given by (5.23), the only difference being the index s in place of r. Therefore limn→∞ g(s) (γ ) = {γ s−1 / (s)}e−γ . The above limiting functions are gamma distributions – (γ ; 1, r) and (γ ; 1, s), respectively – which, however, represent the limit of a function of the relevant order statistics and not of the order statistics themselves. When F(x) is given, it may sometimes be possible to obtain the explicit inverse relation, x = F −1 (γ /n) or x = F −1 (1 − γ /n) as appropriate, but these cases are rather rare. It is worth noting, nonetheless, that considerable work has been done in this direction and it has been found that the limiting distributions of (appropriately standardized) extreme statistics are only of three types, often denoted as Types I, II and III or EV1, EV2 and EV3 – where EV is the acronym for extreme values. Convergence to type I, II or III depends essentially on the ‘tail’ of the underlying distribution F(x) and the rate of convergence is generally rather slow. For more details on this rich and interesting topic the interested reader can refer to [5, 10, 11, 22]. We close this section with two additional comments relevant to the above discussion. First, the sample counterpart of the p-quantile ζp – which is deﬁned implicitly by the equation F(ζp ) = p(0 < p < 1) – is a mid-term order statistic and therefore it is asymptotically normal. It can be shown that its mean and variance are, respectively, ζp and p(1 − p) nf 2 (ζp )

(5.25)

In particular, since ζ1/2 deﬁnes the median of the population, its sample counterpart – X[n/2]+1 if n is odd or any value between X[n/2] and X[n/2]+1 if n is even – is asymptotically normal with mean ζ1/2 and variance {4nf 2 (ζ1/2 )}−1 .

180 Mathematical statistics If, moreover, the parent r.v. is normal with parameters µ, σ 2 , then the sample median is asymptotically normal with parameters µ, (2n)−1 π σ 2 . The second comment refers to the extremal variables X(r) and X(n−s+1) ; if one considers their joint distribution it can be shown that they are asymptotically independent. Both results cited in these comments can be found in [3] or [19].

5.4

Point estimation

As stated at the beginning of this chapter, experimental data are a means to an end: to draw inferences on a population when, for whatever reason, it is not possible to examine the population in its entirety. Clearly, the type of inference – and therefore the desired ﬁnal information – depends on the problem at hand. Nonetheless, some classes of problems are frequently encountered in practice and speciﬁc statistical methods have been devised to address them. Here we consider the parametric model of eq. (5.1) with the aim of ‘estimating’ one or more unknown population parameters. This is one of the typical inference problems and we can choose to give our estimate in one of two distinct forms: (i) by assigning a speciﬁc value to the unknown parameter or (ii) by specifying an interval which – with a given level of conﬁdence – includes the ‘true’ value of the parameter. One speaks of ‘point estimation’ in case (i) and of ‘interval estimation’ in case (ii) and it is understood that (i) and (ii) refer to each one of the unknown parameters when these are more than one. Point estimation is the subject of this and the following sections (Section 5.5 included). Given a sample X1 , . . . , Xn we have considered in the previous sections a number of sample characteristics: each one has the form of a function T(X) = T(X1 , . . . , Xn ) and is a random variable which takes on the value t = T(x) = T(x1 , . . . , xn ) after the experiment has been performed and we have obtained the realization x1 , . . . , xn . If, moreover, T(X) contains no unknown quantities it is generally called a statistic. Intuitively, one would think of estimating an unknown population parameter by using the corresponding statistic so that, for instance, we could use M and S2 as estimators of the population mean and variance µ and σ 2 , respectively. As reasonable as this may sound, things are not always so clear-cut because other statistics can be used for the same purpose and, a priori, there seems to be no reason why M and S2 should be preferred. In order to motivate our choice even in more complex cases, we must ﬁrst try to evaluate the ‘goodness’ of estimators. Let us call θ the unknown parameter to be estimated and let T(X) – or, often, Tn or simply T, implicitly implying the dependence on the sample size – be the statistic used to estimate it. A ﬁrst desirable property for T is that E(T) = θ

for all θ ∈

(5.26)

Preliminary ideas and notions

181

which, in words, is phrased by saying that T is an unbiased estimator (often we will write UE for short) of θ. Note that, strictly, one should write Eθ (T) = θ because the Lebesgue–Stieltjies integral deﬁning the expectation is an integral in dF(x; θ ). This fact, however, is often tacitly assumed for parametric models of the type (5.1). Deﬁning the bias of Tn as b = E(Tn ) − θ it is obvious that Tn is unbiased whenever b = 0. Note that eq. (5.26) does not imply t = θ for every realization of the sample; in fact some realizations will give t − θ > 0 and some others will result in t − θ < 0, however, on average, eq. (5.26) guarantees that there is no systematic error in the evaluation of θ. Also, if g is an arbitrary non-linear function, eq. (5.26) does not imply E[g(Tn )] = √ g(θ ), this meaning, for example, that if Tn is an UE of σ 2 not necessarily Tn is an UE of the standard deviation σ . Besides the bias, another measure of ‘distance’ from the true value is the mean square error (of Tn ), deﬁned as Mse(Tn ) = E[(Tn − θ )2 ]

(5.27a)

which, using the identity T − θ = (T − E(T)) + (E(T) − θ ) = (T − E(T)) + b, can be expressed as Mse(T) = Var(T) + b2

(5.27b)

where E[(T −E(T))2 ] = Var(T) by deﬁnition. Equation (5.27b), in addition, shows that the mean square error of an UE coincides with its variance. So, between any two estimators, say T, T , of the same parameter θ, it seems logical to prefer T if Mse(T) < Mse(T ) for all θ ∈ . If, as it is often the case, we limit our choice to the class of UEs – let us denote this class by u(θ) – the ‘best’ estimator will be the one with minimum variance for all θ ∈ . This minimum-variance-unbiased-estimator (MVUE) T¯ is often called an efﬁcient (or optimum) estimator of θ and satisﬁes the condition ¯ = min {Var(T)} Var(T) T∈u(θ )

for all θ ∈ .

(5.28)

although the concepts are sometimes distinguished because the estimator with minimum variance among all possible estimators (of a given parameter) may not be unbiased. Clearly, one can also speak of relative efﬁciency and compare two estimators on the basis of the ratio of their variances by saying that – given T, T ∈ u(θ) – T is more efﬁcient than T if Var(T) < Var(T ) for all θ ∈ . In this regard, however, it should be noted that it may happen that Var(T) < Var(T ) for some values of θ but Var(T ) < Var(T) for other values of θ. Since the inequality must hold uniformly in θ – that is, for all θ ∈ – and θ is unknown, no efﬁciency comparison can be made in these cases. The same

182 Mathematical statistics consideration applies to efﬁcient estimators and (5.28) may hold for, say, T1 for some θ and T2 for some other θ. Then, efﬁciency is not enough to compare estimators. Within the class u(θ ), the following results hold: Proposition 5.4 If T1 , T2 ∈ u(θ ) are two efﬁcient estimators, then T1 = T2 where the equality T1 = T2 is understood in a probability sense, that is, if T1 and T2 satisfy eq. (5.28) then Pθ (X ∈ {x : T1 (x) = T2 (x)}) = 0 for all θ ∈ . In other words, an efﬁcient estimator, when it exists, is unique. The following proposition, on the other hand, states that efﬁciency is linear: Proposition 5.5 If T1 , T2 , respectively, are efﬁcient estimators of θ1 , θ2 , then a1 T1 + a2 T2 is an efﬁcient estimator of a1 θ1 + a2 θ2 for all a1 , a2 ∈ R. Both proofs can be found in Ref. [19]. A ﬁnal remark is in order: in some cases an UE may not exist or, in other cases, a slightly biased estimator Tb can be preferred to an unbiased one T if Mse(Tb ) < Mse(T) = Var(T) for all θ ∈ . Other desirable properties of estimators consider their behaviour as n → ∞ and not, as above, by regarding the sample size as ﬁxed. These properties are called asymptotic and one says, for instance, that an estimator T is asymptotically unbiased if lim E(Tn ) = θ

n→∞

(5.29)

or equivalently limn→∞ bn = 0, where we write bn because the bias generally depends on the sample size. Clearly, an UE is asymptotically unbiased while the reverse, however, is not true in general. Another asymptotic property is as follows: an estimator Tn of θ is consistent if limn→∞ P(|Tn − θ| < ε) = 1 for all ε > 0, that is, if (see Section 4.3) Tn → θ[P]

(5.30)

Some authors speak of weakly consistent estimator in this case and use the adjective ‘strong’ if Tn → θ [a.s.] or, sometimes, if Tn → θ [M2 ]. In any case (see Propositions 4.6 and 4.8) strong consistency implies weak consistency and, in most cases, the deﬁnition of ‘consistent’ is understood in the sense of eq. (5.30). A useful sufﬁcient condition to determine consistency is given by Proposition 5.6 Tn is a consistent estimator if (a) it is asymptotically unbiased and (b) limn→∞ Var(Tn ) = 0. In fact, if (a) and (b) hold then eqs (5.27b) and (5.27a) imply limn→∞ E[(Tn − θ )2 ] = 0, that is, Tn → θ[M2 ] and therefore Tn → θ[P]. It is evident that

Preliminary ideas and notions

183

(b) only must hold if Tn is unbiased. Note also that requirements (a) and (b) are not necessary; in fact it can be shown that there are consistent estimators whose variance is not ﬁnite. Owing to the properties of P-convergence we have also: Proposition 5.7 If Tn is a consistent estimator of θ and g is a continuous function, then g(Tn ) is a consistent estimator of g(θ ). All the deﬁnitions and considerations above extend readily to the case of more than one unknown parameter θ1 , θ2 , . . . , θk (k > 1) which, as noted in Section 5.2, can be considered as a k-dimensional vector. To end this section, a ﬁnal word of caution on asymptotic properties of estimators is not out of place. In practical cases, these properties provide valid criteria of judgement for large samples but lose their meaning for small samples. Unfortunately, the notions of ‘small’ or ‘large’ samples often depend on the problem at hand and cannot be made more precise without considering speciﬁc cases. A general rule of thumb requires n > 30 in order to be able to speak of ‘large samples’; caution, however, must be exercised because the exceptions to this ‘rule’ are not rare. Example 5.4(a) Equation (5.9) and the ﬁrst of (5.15) show that the statistics M and Ak are UEs of the population parameters µ and αk , respectively. Equation (5.12), however, shows that S2 is a biased estimator of σ 2 , the bias being b = −σ 2 /n. Since b → 0 as n → ∞, S2 is an asymptotically unbiased estimator of σ 2 (also, it is consistent because it satisﬁes the requirements of Proposition 5.6). For ﬁnite samples, nonetheless, the bias can be removed by considering the estimator S¯ 2 =

n 1 S2 (Xi − M)2 = n−1 n−1 n

(5.31a)

i=1

which satisﬁes E(S¯ 2 ) = σ 2 (note that some authors use the name ‘sample variance’ to denote the statistic S¯ 2 ). Also, for a normal population we have from eqs (5.14b) and (5.31a) Var(S¯ 2 ) =

2σ 4 n−1

(5.31b)

The procedure of bias removal shown above can be generalized to all cases in which E(T) = c+dθ – where c, d are two known constants – by deﬁning T¯ = (T − c)/d. Then, the statistic T¯ is an UE of θ. Another example of this type is the statistic C3 as an estimator of µ3 because E(C3 ) = n−2 (n − 1)(n − 2)µ3 (the reader is invited to check this result).

184 Mathematical statistics In cases where the population mean µ is known the quantity 1 Sˆ 2 = (Xi − µ)2 n n

(5.32)

i=1

is a statistic in its own right. It is left to the reader to show that (i) Sˆ 2 is an UE of σ 2 and (ii) Var(Sˆ 2 ) = n−1 (µ4 − σ 4 ) = 2n−1 σ 4 where the last equality is due to the fact that, for a normal r.v., µ4 = 3σ 4 (see eq. (2.44d)). Example 5.4(b) From the considerations above it is evident that, in general, there exist many unbiased estimators of a given parameter. As a further example of this, it is easy to show that any linear combination Tˆ = ni=1 ci Xi such that c1 +c2 +· · ·+cn = 1 is an UE of µ. Turning to its variance, however, ˆ = σ 2 c2 and since we have Var(T) i i i

ci2

=

i

1 ci − n

2 +

1 n

it follows that the minimum value of the sum i ci2 occurs when ci = 1/n for all i = 1, . . . , n. Consequently, the sample mean M is the most efﬁcient ˆ If, in particular, the sample comes among all estimators (of µ) of the form T. from a normal population N(µ, σ 2 ), we noted at the end of the preceding section that the sample median – let us denote it by Z – is asymptotically normal with parameters µ and (2n)−1 π σ 2 . For large samples, therefore, Z can be chosen as an estimator of µ but since (Proposition 5.1b) Var(M) < Var(Z), the sample mean is more efﬁcient than Z. We open here a short parenthesis: the fact that the sample median is less efﬁcient than M should not lead the analyst to discard Z altogether as an estimator of µ. In fact, this statistic is much more robust than M and this quality is highly desirable in practice when the data may be contaminated by ‘outliers’. We do not enter in any detail here but we only say that ‘robust’ in this context means that Z, as an estimator of the mean, is much less sensitive than M to the presence of outliers, where the term ‘outlier’ denotes an unexpectedly high or low value which, at ﬁrst sight, does not seem to belong to the sample. As a matter of fact, this is often the case because outliers are generally due to recording, transmission or copying errors; in some cases, however, they may be true data of exceptional events. The interested reader can refer, for example, to Chapter 16 of [27]. Example 5.4(c) Turning brieﬂy to asymptotic properties it is immediate to show, for instance, that M and Ak are consistent estimators of µ and αk , respectively. In fact, they are unbiased and their variance – see eq. (5.10) and the second of (5.15) – satisfy condition (b) of Proposition 5.6. Also, having

Preliminary ideas and notions

185

σ2

S2

already noted that is an asymptotically unbiased estimator of we can use eq. (5.14) to determine that – if µ4 exists – then Var(S2 ) → 0 as n → ∞ and therefore S2 is a consistent estimator of σ 2 by virtue of Proposition 5.6. Example 5.4(d) As a ﬁnal example, let us suppose that the parent r.v. X of the sample is distributed according to a Cauchy pdf of the form f (x; θ ) =

1 π[1 + (x − θ )2 ]

(5.33)

which represents our parametric model. Suppose further that we consider the sample mean M as an estimator of θ. Now, using characteristic functions it is not difﬁcult to show that M has the same distribution as X and therefore the probability P(|M − θ| ≥ ε) – being the same for all n – cannot tend to zero as n → ∞. The conclusion is that M is not a consistent estimator of θ. 5.4.1

Cramer–Rao inequality

In the preceding section we deﬁned the relative efﬁciency of estimators by restricting our attention to the class u(θ ) of UEs. Within this class, the requirement of minimum variance – see eq. (5.28) – is the property of interest. Suppose, however, that somehow (we will have to say more about this later) we can ﬁnd some unbiased estimators of a given parameter θ. Among these estimators, we can select the most efﬁcient, but how do we know that there are no more efﬁcient ones? In many cases, the Cramer–Rao inequality can answer this question. In fact, provided that some ‘regularity conditions’ are satisﬁed, it turns out that the variance of UEs is bounded from below; if we ﬁnd an estimator whose variance equals this lower bound then we also know that this estimator – in the terms speciﬁed by Proposition 5.4 – is unique. In order to keep things relatively simple, we consider the one-dimensional continuous case and denote by f (x; θ ) the pdf of the parent r.v. X of the sample (X1 , . . . , Xn ). Then the so-called likelihood function L(x; θ ) = L(x1 , . . . , xn ; θ ) =

n

f (xi ; θ )

(5.34)

i=1

is the pdf of the sample. We assume the following regularity conditions: (a) the set {x : f (x; θ ) > 0} – that is, in mathematical terminology, the support of the pdfs f (x; θ ) – does not depend on θ; (b) thefunction f (x; θ) is differentiable with respect to θ; ∂ ∂ (c) ∂θ f (x; θ ) dx f (x; θ ) dx = ∂θ ∂ ∂ (d) ∂θ T(x)L(x; θ ) dx = T(x) ∂θ L(x; θ ) dx where all (Lebesgue) integrals are on all space (R in (c) and Rn in (d))

186 Mathematical statistics (e) E[U 2 (X; θ )] < ∞ where the ‘score’ or ‘contribution’ function U(X; θ ) is deﬁned as ∂ ∂ ln L(X; θ ) = ln f (Xi ; θ ) ∂θ ∂θ n

U(X; θ ) =

(5.35)

i=1

and the second equality descends from (5.34). Proposition 5.8 (Cramer–Rao inequality) ditions, let T = T(X) ∈ u(θ ). Then Var(T) ≥

1 E[U 2 (X; θ )]

=

Under the above regularity con-

1 In (θ )

(5.36)

where the function E[U 2 (X; θ )], being important in its own right, is denoted by In (θ) and called Fisher’s information (on θ) contained in the sample X. As a preliminary result, note that E[U(X; θ )] = 0

(5.37)

In fact, since f (X; θ ) = f (X1 ; θ ) = · · · = f (Xn ; θ ), from (5.35) it follows

∂ ∂ E[U(X; θ )] = ln f (Xi ; θ ) = nE ln f (X; θ ) E ∂θ ∂θ i ∂ ∂ = n f (x; θ ) [ln f (x; θ )] dx = n f (x; θ ) dx ∂θ ∂θ ∂ =n f (x; θ ) dx = 0 ∂θ where we used the relation ∂ ln f /∂θ = (1/f )(∂f /∂θ ) in thefourth equality, condition (c) in the ﬁfth and the last equality holds because f (x; θ ) dx = 1. Now, in order to prove eq. (5.36) we apply Cauchy–Schwarz inequality (eq. (3.21)) to the variables (T(X) − θ ) and U(X; θ ) E2 [(T − θ )U] ≤ E[(T − θ )2 ]E(U 2 ) = Var(T)E(U 2 )

(5.38)

Since E[(T − θ )U] = E(TU) − θE(U) = E(TU), we use the relation ∂ ln L/∂θ = (1/L)(∂L/∂θ ) and condition (d) to get ∂ ∂L(x; θ ) E(TU) = T(x)L(x; θ ) [ln L(x; θ )] dx = T(x) dx ∂θ ∂θ ∂ ∂ ∂ = E(T) = θ =1 T(x)L(x; θ ) dx = ∂θ ∂θ ∂θ

Preliminary ideas and notions

187

so that the l.h.s of (5.38) is unity and Cramer–Rao inequality follows. Furthermore, by considering the explicit form of U of eq. (5.35) we get ⎧ 2 ⎫

2 6 ⎬ ⎨ ∂ ∂ 2 In (θ)= E(U )= E = ln f (Xi ; θ ) ln f (Xi ; θ ) E ⎭ ⎩ ∂θ ∂θ i i

( ' ∂ ∂ + ln f (Xi ; θ ) ln f (Xj ; θ ) E ∂θ ∂θ i =j

2 6 ∂ = nI(θ ) = nE ln f (X; θ ) ∂θ where (i) the sum on i = j is zero because of independence and of eq. (5.37) and (ii) I(θ) = E{(∂ ln f (X; θ )/∂θ )2 } is called Fisher’s information and is the amount of information contained in one observation; the fact that In (θ) = nI(θ) means that the information of the sample is proportional to the sample size. In the light of these considerations we can rewrite (5.36) as Var(T) ≥

1 nI(θ )

(5.39)

If, in addition, f is twice θ-differentiable and we can interchange the signs of integration and derivative twice we have yet another form of the inequality. In fact, while proving eq. (5.37) we showed that (∂f /∂θ ) dx = 0; under the additional assumptions, we can differentiate with respect to θ to get 0=

∂ 2f dx = ∂θ 2

1 f

∂ 2f ∂θ 2

1 ∂ 2f f dx = E f ∂θ 2

and since we can use this last result to obtain

∂ 2 ln f ∂ 1 ∂f 1 ∂f 2 E = E = −E ∂θ f ∂θ ∂θ 2 f 2 ∂θ

∂ ln f 2 1 ∂ 2f +E = −E = −I(θ ) f ∂θ 2 ∂θ Cramer–Rao inequality can be written as "

∂ 2 f (X; θ ) Var(T) ≥ − nE ∂θ 2

#−1 (5.40)

188 Mathematical statistics A few remarks are in order: (1) the equal sign in (5.36) holds if and only if the two r.v. in Cauchy–Schwarz inequality are linearly related, that is, when T(X) − θ = a(θ )U(X; θ )

(5.41)

where a(θ) is some function of θ. (2) if T is an UE of a (differentiable) function τ (θ) of θ, the numerator of Cramer–Rao inequality becomes {τ (θ )}2 and eq. (5.41) becomes T − τ (θ ) = a(θ )U. Whenever this relation holds, then Var(T) = a(θ )E(TU). Then, since the θ-derivative of τ (θ) = E(T) = T(x)L(x; q) dx is τ (θ ) = E(TU), it follows that Var(T) = a(θ )τ (θ )

(5.42)

which reduces to Var(T) = a(θ ) if, as we considered above, τ (θ) = θ. (3) Cramer–Rao inequality establishes a lower bound for the variance of an UE; this does not imply that an estimator with such minimum variance exists (when this is not the case, one may use Bhattacharya’s inequality; for more details see for instance Ref. [17] or [19]). (4) the ratio between the lower bound and Var(T) is called efﬁciency of the estimator and denoted by eT , that is, eT =

1 1 = In (θ)Var(T) nI(θ )Var(T)

(5.43)

where 0 ≤ eT ≤ 1 and eT = 1 indicates a MVUE estimator. (5) The discrete case, with only minor modiﬁcations is analogous to the continuous one. Example 5.5(a) Consider a sample from a normal distribution with unknown mean θ = µ and known variance σ 2 . All regularity conditions are met and f (x; θ ) is given by eq. (2.29a). Then ∂ ln f (x; µ)/∂µ = (x − µ)/σ 2 and Fisher’s information is * I(µ) = E

∂ ln f (x; µ) ∂µ

+2 =

1 1 E[(X − µ)2 ] = 2 σ4 σ

which, as expected, implies that a smaller variance corresponds to a higher information. Now, considering M as an estimator of µ and knowing that (eq. (5.10)) Var(M) = σ 2 /n we get eM = 1; therefore M is a MVUE estimator of µ. Also, eq. (5.41) must hold. In fact, using the expression of f (Xi ; θ )

Preliminary ideas and notions

189

pertinent to our case, we get from eq. (5.35)

U(X; µ) =

n n 1 (Xi − µ) = 2 (M − µ) σ2 σ i=1

which, in fact, is eq. (5.41) where T − θ = M − µ and a(θ ) = σ 2 /n. Note that, in agreement with eq. (5.42), a(θ ) = Var(M). Example 5.5(b) Turning to a discrete case, consider a sample from a parent Poisson r.v. X with unknown parameter θ = λ. From the pmf of eq. (4.1) we get the Fisher’s information I(λ) = λ−1 and since Var(X) = λ implies Var(M) = λ/n, we have again eM = 1. (Incidentally, it is not out of place to point out that examples (a) and (b) must not lead to the (wrong) conclusion that M – although always unbiased – is always an efﬁcient estimator of the mean.) Example 5.5(c) Exponential Models. An important class of parametric models has the general form f (x; θ ) = exp{A(θ )B(x) + C(θ ) + D(x)}

(5.44)

and is called exponential. Not all exponential models satisfy the regularity conditions, but for the ones that do the following considerations apply. Denoting by a prime the derivative with respect to θ, the score function is easily obtained as U(X; θ ) = A (θ )

n i=1

"

C (θ ) 1 B(Xi ) + nC (θ ) = nA (θ ) B(Xi ) + n A (θ ) n

#

i=1

which corresponds to eq. (5.41) once we set (see also remark (2)) T(X) = n−1

n

B(Xi )

i=1

τ (θ) = −C (θ )/A (θ )

(5.45)

a(θ) = [nA (θ )]−1 from which it follows that for the exponential class the statistic T(X) is an efﬁcient estimator of τ (θ ), where T(X) and τ (θ ) are given by the ﬁrst and

190 Mathematical statistics second parts of eq. (5.45). This, by eq. (5.42), implies Var(T) =

τ (θ ) nA (θ )

(5.46)

Also, only a small effort is required to show that I(θ) = τ (θ )A (θ )

(5.47)

By appropriately identifying the functions A, B, C and D, many practical models are, as a matter of fact, exponential. Examples (a) and (b), for instance, are two such cases. In fact, if we set A(θ ) = θ/σ 2 , B(x) = x, C(θ) = −θ 2 /2σ 2 and D(x) = −x2 /2σ 2 we ﬁnd example (a) and, as above, we determine that M is an efﬁcient estimator of µ with Var(M) = σ 2 /n. The Poisson example of case (b), on the other hand, is obtained by setting A(θ) = ln θ, B(x) = x, C(θ ) = −θ and D(x) = − ln x!. Example 5.5(d) As a further special case of exponential model, the reader is invited to consider a sample from a normal population with known mean 2 = σ 2 . By setting A(θ ) = −1/2θ 2 , B(x) = µ and unknown variance θ√ 2 (x − µ) and C(θ ) = − ln(θ 2π ) it turns out that T(X) = n−1 (Xi − µ)2 is an efﬁcient estimator of τ (θ ) = θ 2 . Also, the reader should check that 2 −1 3 −1 2 eq. (5.41) for this case is n i (Xi − µ) − θ = n θ U(X; θ ) and that 4 Var(T) = 2θ /n, in agreement with result (ii) of Example 5.4(a). The above examples show that for large samples the order of the variance of UEs is n−1 . This, as a matter of fact, is a general rule which applies to regular models. It is worth pointing out that in some cases of non-regular models it is possible to ﬁnd UEs whose variance decreases more quickly than n−1 as n increases – that is, we can ﬁnd UEs with variances smaller than the Cramer–Rao limit. Examples of these ‘superefﬁcient’ estimators can be found, for instance, in Chapter 32 of [3] or in Chapter 2 of [19]. In closing this section, we brieﬂy outline the case of more than one parameter, let us say k, so that q = (θ1 , θ2 , . . . , θk )T is a k-dimensional vector. Then, the score function is itself a vector U = (U1 , . . . , Uk )T where Ui (X; q) = ∂ ln L(X; q)/∂θi and one can form the k × k information matrix of the sample as In (q) = E(UUT ) = nI(q)

(5.48)

(the second equality is the vector counterpart of the one-dimensional relation In (θ) = nI(θ), valid in our experiment of repeated independent trials). The ijth element Iij (q) of I(q) – which, in turn, is the information matrix of one

Preliminary ideas and notions

191

observation – is given by (i, j = 1, 2, . . . , k)

∂ ln f (X; q) ∂ ln f (X; q) Iij (q) = E ∂θi ∂θj

∂ 2 ln f (X; q) = −E ∂θi ∂θj

(5.49)

and the last relation holds if f (x; q) is twice differentiable with respect to the parameters θ1 , . . . , θk . Clearly, both In and I are symmetric, that is, In = InT and I = IT . Given these preliminary notions, let T(X) be an unbiased estimator of some function τ (q) = τ (θ1 , . . . , θk ) of the unknown parameters; then the Cramer–Rao inequality is now written Var(T) ≥ dT In−1 d =

1 T −1 d I d n

(5.50)

where d = d(q) is the vector of derivatives d(q) = (∂τ/∂θ1 , . . . , ∂τ/∂θk )T and, similarly to eq. (5.41), the equality holds if and only if T(X) − τ (q) = [a(q)]T U(X; q)

(5.51)

for some vector function a = (a1 , . . . , ak )T of the parameters (note that, in general, ai = ai (q) for all i = 1, . . . , k). Moreover, as in the one-dimensional case, one calls efﬁcient an estimator of τ (q) whose variance coincides with the r.h.s. of (5.50). Finally, since it is evident that eq. (5.50) holds only if In (q) (and therefore I(q)) is non-singular for all q ∈ , this assumption is generally added to the other deﬁning conditions of regularity. 5.4.2

Sufﬁciency and completeness of estimators

In order to evaluate the ‘goodness’ of an estimator, another desirable property – besides the ones considered so far – is sufﬁciency. The deﬁnition is: given an unknown parameter θ, an estimator T(X) of θ is sufﬁcient (or exhaustive for some authors) if the conditional likelihood L(x|T = t; θ ) does not depend on θ. Equivalently, T is sufﬁcient if the conditional probability Pθ (X ∈ A|T = t) does not depend on θ for any event A ⊂ . This deﬁnition is not self-evident and some further comments may help. In essence, sufﬁciency requires that the values t = T(x1 , . . . , xn ) taken on by the statistic T must contain all the information we can get on θ. In other words, suppose that two realizations of the sample x and x both lead to the value t = T(x) = T(x ). If the function L(x|T = t; θ ) depended on θ then we would have, say, L(x|T = t) > L(x |T = t) for θ ∈ 1 and L(x|T = t) < L(x |T = t) for θ ∈ 2 , where 1 ∪ 2 = and 1 ∩ 2 = ∅ (i.e. the sets 1 , 2 form a partition of the parameter space ). Therefore, knowing which one of the two realization has occurred provides more information than just the fact of knowing that T = t. So, for instance, if x has occurred,

192 Mathematical statistics we would tend to think that, preferably, θ ∈ 1 . If, on the other hand L(x|T = t) = L(x |T = t) for all θ ∈ then the speciﬁc realization of the sample leading to T = t is irrelevant and – for a ﬁxed sample size n – the equality T = t summarizes all that we can know in order to estimate θ. This is why T is a ‘sufﬁcient’ estimator of θ. In practice, it may be difﬁcult to determine sufﬁciency just by using the deﬁnition above. Often, an easier way to do it is to use Neyman’s theorem (which some authors give as the deﬁnition of sufﬁciency) Proposition 5.9(a) (Neyman’s factorization theorem) A statistic T(X) is sufﬁcient for θ if and only if the likelihood function can be factorized into the product of two functions g(T(x); θ ) and h(x), that is, L(x; θ ) = g(T(x); θ )h(x)

(5.52)

(where it should be noted that the factor g depends on x only through T(x)). In fact, since L(x|t; θ ) =

Pθ (X = x ∩ T = t) L(x; θ ) = L(x ; θ ) Pθ (T = t)

(the sum at the denominator is over all realizations x giving T = t) if we assume that the factorization (5.52) holds we get L(x|t; θ ) = h(x)/ h(x ) and therefore, according to the deﬁnition above, T is sufﬁcient (if x is such that T(x) = t then L(x|t; θ ) = 0; consequently L(x|t; θ ) does not depend on θ for any realization of the sample). The proof of the reverse statement – that is, if L(x|t; θ ) does not depend on θ then eq. (5.52) holds – is left to the reader. Example 5.6(a) Let X be a sample from a Poisson variable (see eq. (4.1)) of unknown parameter θ. Then L(x; θ ) =

n

i=1

e−θ

θ xi θ x1 +···+xn = e−nθ xi ! x1 ! · · · xn !

(5.53)

and eq. (5.52) holds with g = e−nθ θ x1 +···+xn and h = (x1 ! · · · xn !)−1 . It follows that the statistic T(X) = X1 + · · · + Xn is a sufﬁcient estimator of θ. Alternatively, in this case we could also use the deﬁnition by noting (Section 4.2) that T is itself a Poisson variable of parameter nθ. So, Pθ (T = t) = {e−nθ (nθ )t }/t! and L(x|t; θ ) is independent on θ because L(x|t; θ ) =

e−nθ θ t t! = t {Pθ (T = t)}x1 ! · · · xn ! n (x1 ! · · · xn !)

Preliminary ideas and notions

193

Example 5.6(b) Let X be a sample from a Gaussian variable with unknown mean µ = θ and known variance σ 2 . Then, deﬁning T(x) = x1 + · · · + xn we have n

1 (xi − θ )2 L(x; θ ) = exp − √ 2σ 2 2π σ i=1 (5.54)

n 1 1 = √ x2i − 2θT(x) + nθ 2 exp − 2 2σ 2π σ i

and since Neyman’s theorem holds by choosing +

*

1 g(T(x); θ ) = exp − 2 (2θT(x) + nθ 2 ) 2σ √ 1 2 −n h(x) = ( 2π σ ) exp − 2 xi 2σ i

the statistic T(X) = X1 + · · · + Xn is a sufﬁcient estimator of θ. A corollary to Proposition 5.9(a) is Proposition 5.10 (i) If the function z is one-to-one and T is sufﬁcient for θ, then Z = z(T) is also a sufﬁcient estimator of θ. Moreover, (ii) Z = z(T) is a sufﬁcient estimator of θˆ = z(θ ). In fact, the relation L(x; θ ) = g(z−1 (Z); θ )h(x) = g1 (Z; θ )h(x) proves part (i) while part (ii) follows easily by also considering the relation θ = z−1 (θˆ ). An immediate consequence of the corollary is that if T(X) = X1 + · · · + Xn is a sufﬁcient estimator for the mean µ of a population, so is M = T/n. At this point, an important observation is that Neyman’s factorization (5.52) implies eq. (5.41) which – as we have determined – characterizes efﬁcient (MVUE) estimators. In other words, this means that the class of sufﬁcient statistics (for the parameter θ) includes the MVUE of θ when this estimator exists (note, however, that sufﬁcient statistics may exist even when there is no MVUE). Moreover, Rao–Blackwell theorem states that the following: Proposition 5.11 (Rao–Blackwell) of a sufﬁcient statistic.

The MVUE, when it exists, is a function

In fact, let X be a sample from a population with an unknown parameter θ, T(X) a sufﬁcient statistic for θ and T1 (X) an arbitrary UE of θ. Then E(T1 |T) (note that in strict symbolism we should write Eθ (T1 |T)) is a function of the form H(T) which takes on the value H(t) = E(T1 |t) when T = t. Since

194 Mathematical statistics T, T1 are random variables in their own right, we can use eq. (3.89a) to get E[H(T)] = E[E(T1 |T)] = E(T1 ) = θ where the last equality holds because T1 is unbiased. The consequence is that H(T) is itself an UE of θ. In addition to this, eq. (3.91) shows that Var(T1 ) = E[Var(T1 |T)] + Var[H(T)] which – since E[Var(T1 |T)] ≥ 0 – implies Var[H(T)] ≤ Var(T1 )

(5.55)

(the equal sign holds if and only if E[Var(T1 |T)] = E{[T1 − E(T1 |T)]2 } = 0, that is, whenever T1 = H(T) – or, more precisely, when P{T1 = H(T)} = 1). At this point one could conclude that H(T) is (i) an UE of θ and (ii) more efﬁcient than T1 . Before doing this, however, one must show that H(T) is a statistic, that is, does not depend on θ. By recalling eq. (3.88) we can write H(t) = E(T1 |t) =

T1 (x)L(x|t; θ ) dx

and note that both L(x|t; θ ) and T1 (x) do not depend on θ because, respectively, T is sufﬁcient and T1 is a statistic. So, H(t) does not depend on θ; moreover, as t varies the r.v. H(T) takes on the values H(t) with a density fT (t) which is itself independent on θ (T is a statistic). Consequently, as desired, H(T) is a statistic. Despite its intrinsic importance, Rao–Blackwell theorem is of little help in explicitly ﬁnding the MVUE (assuming that it exists). In fact, given a sufﬁcient and an unbiased estimator, T and T1 respectively, we can construct the UE H(T) which – although more efﬁcient than T1 – may not be the MVUE of θ. In principle, by using H(T) and another sufﬁcient statistic, we expect to be able to ﬁnd an even more efﬁcient (than H(T)) UE. However, if the original sufﬁcient statistic is complete (see deﬁnition below), it turns out that H(T) is the MVUE of the parameter θ. This is stated in the following proposition: Proposition 5.12 (Lehmann–Scheffé theorem) Let T(X) be a sufﬁcient and complete statistic for θ and T1 (X) an UE of θ. Then H(T) = E(T1 |T) is the efﬁcient estimator of θ. Before showing why this is so, we give the deﬁnition of completeness: a sufﬁcient statistic T is complete if for any (bounded) function ϕ(T) the relation Eθ [ϕ(T)] = 0

for all θ ∈

implies ϕ(t) = 0 for almost all values t = T(x) (the term ‘almost all’ refers to the measure Pθ and indicates that Pθ {ϕ(T(x)) = 0} = 1 for all θ ∈ ).

Preliminary ideas and notions

195

Returning to Proposition 5.12, assume that there exists another UE K(T) depending on T. Deﬁning L(T) = H(T) − K(T), we have E[L(T)] = θ − θ = 0 for all θ and this, by completeness, implies H(t) = K(t) a.e. which, in turn, shows that H(T) is the unique UE depending on T. Let now T˜ be ˜ an arbitrary UE. By virtue of the considerations above J(T) = E(T|T) is ˜ ˜ unbiased, Var[J(T)] ≤ Var(T) and the equality holds iff T = J(T). Since H(T) is the only UE depending on T, then we must have J(T) = H(T) and this proves the theorem. At this point, two closing remarks on sufﬁciency are worthy of mention. First we outline the generalization to the case of k unknown parameters. In this case the following deﬁnition applies: the vector T = (T1 , . . . , Tk ) is called a (jointly) sufﬁcient statistic for q = (θ1 , . . . , θk ) if the function L(x|t1 , . . . , tk ; q) does not depend on q. Neyman’s theorem, on the other hand, becomes: Proposition 5.9(b) The k-dimensional statistic T(X) = (T1 (X), . . . , Tk (X)) is (jointly) sufﬁcient for q = (θ1 , . . . , θk ) if and only if the likelihood function can be expressed as the product L(x; q) = g(T1 (x), . . . , Tk (x); q)h(x)

(5.56)

So, for example, it is easy to show that T = (T1 , T2 ) – where T1 = i Xi 2 and T2 = i Xi – is a sufﬁcient statistic for the two-dimensional Gaussian model with unknown mean and variance. Using the sufﬁcient statistic T we can then construct the well-known estimators M = n−1 T1 and S¯ 2 = (n − 1)−1 [T2 − n−1 T12 ] (see eq. (5.31)) of µ and σ 2 . The second and ﬁnal remark may appear rather obvious at ﬁrst glance but – we believe – deserves to be stated explicitly: sufﬁciency depends on the adopted statistical model. In other words, if the model is changed, a given sufﬁcient statistic may no longer be sufﬁcient in the new model. As a consequence, we should never discard the raw data and replace them with sufﬁcient statistics. In fact, although the main advantage of sufﬁcient statistics is to reduce the dimensionality of the sample without losing any information on the unknown parameter(s), it should also be kept in mind that the sample itself X = (X1 , . . . , Xn ) is always a sufﬁcient statistic irrespective of the adopted statistical model. Consequently – since the model may always be changed in the light of new evidence or of new assumptions – it is always good practice to preserve the original data. As an example, consider a sequence of binomial trials with unknown probability of success θ = p. The order of successes and failures is clearly unimportant in a model of independent trials and the sufﬁcient statistic T = X1 + · · · + Xn is equivalent to the sample as far as the estimation of θ is concerned. However, it can be shown [8] that it is not so if a new model of dependent trials is postulated.

196 Mathematical statistics (Incidentally, under the assumption of binomial independent – that is, Bernoulli – trials, the reader is invited to show that T = X1 + · · · + Xn is, indeed, a sufﬁcient statistic.)

5.5

Maximum likelihood estimates and some remarks on other estimation methods

In regard to point estimation, not much has been said so far on the way in which we can ﬁnd ‘good’ estimators although, in the preceding section, we have implicitly given a method of ﬁnding the MVUE of a parameter θ by using an UE T1 , a sufﬁcient complete statistic T and calculating the conditional expectation E(T1 |T). This procedure, however, often involves computational difﬁculties and is seldom used in practice. Other methods, in fact, have been devised and the most popular by far is the so-called ‘method of maximum likelihood’, introduced by Fisher in 1912 (although the deﬁnition of likelihood, also due to Fisher, appeared later). Before considering this, however, it is worth spending a few words on other methods with the main intention of simply illustrating – without any claim of completeness – other approaches to the problem. One of the oldest estimation procedures is Pearson’s ‘methods of moments’ and consists in equating an appropriate number of sample moments to the corresponding population moments which, in turn, depend on the unknown parameters. By considering as many moments as there are parameters, say k, one solves the resulting equations for θ1 , . . . , θk thus obtaining the desired estimates. In mathematical terms, if j = 1, . . . , k and aj = Aj (x) are the sample moments of the observed realization x = (x1 , . . . , xn ), one must solve the set of equations αj (θ1 , . . . , θk ) = aj ,

j = 1, 2, . . . , k

(5.57a)

whose result is in the form θj = tj (a1 , . . . , ak ),

j = 1, 2, . . . , k

(5.57b)

where the tj ’s – that is, the values taken on by the estimators Tj ’s at X = x – are obtained as functions of the sample moments. Recalling the developments of Section 5.3.1, this last observation on the Tj implies, under fairly general conditions, two desirable properties: for large samples the Tj are (i) consistent and (ii) asymptotically normal. Often, however, they are biased and their efﬁciency, as Fisher himself has pointed out [9] may be rather poor. For small samples, moreover, it should be kept in mind that sample moments may signiﬁcantly differ from their population counterparts, thus leading to poor estimates. This is especially true if higher-order moments must be used because in these cases n < 100 is generally considered a small sample.

Preliminary ideas and notions

197

As an example of the method, consider a sample X from a population with unknown mean µ = α1 and variance σ 2 = α2 − α12 . Equations (5.57a) and (5.57b) are simply αj = aj (j = 1, 2), and since a1 = m = n−1 i xi and a2 = n−1 i x2i , we get t1 = m and t2 = a2 − m2 = n−1 i (xi − m)2 . The desired estimators are therefore T1 = M T 2 = A2 − M 2 =

1 (Xi − M)2 n i

where we already know (eq. (5.12)) that T2 = S2 is a biased estimator of σ 2 . In all, however, the method has the advantage of simplicity and the ‘moments-estimates’ can be used as a ﬁrst approximation in view of a more reﬁned analysis. A second method of estimation is based on Bayes’ formula (eq. (3.79) in the continuous case). If we consider the unknown parameter θ as a value taken on by a r.v. Q with pdf fQ (θ ) – which, somehow, must be known by some prior information and for this reason is called ‘a priori’ density – Bayes’ formula yields (taking eq. (3.80b) into account) f (θ|x) = ∞

f (x|θ )fQ (θ )

−∞ f (x|θ )fQ (θ ) dθ

(5.58)

Then, by deﬁning Bayes’ estimator (of θ) as TB ≡ E(Q|X), its value tB corresponding to the realization x is taken as the estimate of θ, that is, ∞ tB = E(Q|X = x) =

θf (θ|x) dθ

(5.59)

−∞

A few additional comments on this method are worthy of mention. First, the function f (x|θ ) at the numerator of (5.58) is just the pdf f (x; θ ) that speciﬁes the statistical model. However, the point of view is different; instead of seeing θ as a deterministic quantity and postulating the existence of a ‘true’ value θ0 which – were it known – would provide the ‘exact’ probabilistic description by means of f (x; θ0 ), the Bayesian approach considers θ as a random variable and writes f (x|θ ) to mean that the realization x is conditioned by the event Q = θ. In this light, the ‘a posteriori’ density f (θ|x) provides information on θ after the realization x has been obtained and consequently we can use it to calculate the quantity E(Q|X = x) which, in turn – being the mean value of Q given that the event X = x has occurred – is a good candidate as an estimate of θ. Nonetheless, a key point of the method is how well we know fQ (θ). This, clearly, depends on the speciﬁc case under study although

198 Mathematical statistics it has been argued that a uniform distribution for Q may be used in cases of no or very little prior information (a form of the so-called ‘principle of indifference’). We do not enter into the details of this debated issue, which is outside the scope of the book, and pass to the main subject of this section: the method of maximum likelihood. Consider the statistical model (5.1) with k unknown parameters q = (θ1 , . . . , θk ). Once a realization of the sample x has been obtained, the likelihood L(x; q) is a function of q only; consequently, we can write L(q) and note that this function expresses the probability (density) of obtaining the result that, in fact, has been obtained, that is, x. In this light it is reasonable to assume as ‘good’ estimates of the unknown parameters the values qˆ = (θˆ1 , . . . , θˆk ) that maximize L(q), that is, ˆ = Max L(q) L(q)

(5.60)

q∈

where it should be noted that the maximum is taken on the parameter space and not on all the possible values that make mathematical sense for L(q). Owing to (5.60), θˆ1 , . . . , θˆk are called maximum likelihood (ML) estimates of θ1 , . . . , θk . As x varies, we will obtain different values of qˆ and this correspondence leads to the deﬁnition of ‘maximum likelihood estimators’ (MLE) as those statistics Tˆ 1 (X), . . . , Tˆ k (X) which, respectively, take on the values θˆ1 , . . . , θˆk when X = x. In practice, the ML estimates are obtained by ﬁnding the maximum of the log-likelihood function l(q) = ln L(q) (which is equivalent to maximizing L(q)), that is, by ﬁrst solving the likelihood equations ∂l(θ1 , . . . , θk ) = 0, ∂θj

j = 1, 2, . . . , k

(5.61)

and then checking which solution is an absolute maximum (in fact, the solutions of eq. (5.61) – if there are any – determine the stationary points of l(q), which can be minima, maxima or saddle points). The whole procedure is generally rather easy if we have one (or two) unknown parameter(s) but it is evident that computational difﬁculties may arise for higher values of k. In these cases one must resort to numerical techniques of solution of eq. (5.61) and the Newton–Raphson iteration method is frequently used for this task. The subject, however, is beyond our scope and the reader interested in computational aspects may refer, for instance, to [29] (Incidentally, in regard to the determination of the maximum among the solutions of (5.61), it may be worth recalling a theorem of analysis which states the following: If l(q) is twice differentiable and is an open set of Rk , a maximum is attained ˆ T H(q)(q ˆ ˆ deﬁned by the Hessian matrix at qˆ if the quadratic form (q − q) − q) 2 H(q) = [∂ l/∂θi ∂θj ] (i, j = 1, . . . , k) is negative deﬁnite).

Preliminary ideas and notions

199

Example 5.7(a) Considering a sequence of n Bernoulli trials, the statistical model is clearly given by (5.2). Then, the ML estimate of the parameter θ = p is easily obtained by ignoring the terms with the factorials (which do not involve θ) and writing l(θ ) = x ln θ + (n − x) ln(1 − θ ). Taking the derivative ∂l x n−x = − =0 ∂θ θ 1−θ we get the solution θˆ = x/n, which is a maximum because ∂ 2 l/∂θ 2 is negˆ Also note that the ML estimate coincides with the ative at the point θ = θ. observed frequency of success. This speciﬁc example is one among many others that, a posteriori, justiﬁes the relative frequency approach to probability discussed in Chapter 1. Example 5.7(b) In the case of a sample from a normal population with unknown mean µ = θ1 and variance σ 2 = θ2 , the reader is invited to ˆ1 = n−1 i xi = m and θˆ2 = determine that the ML estimates are θ n−1 i (xi − m)2 = s2 so that the MLE are M and S2 , respectively. The examples above do not do justice to the ML method because the reader can easily check that the method of moments yields the same estimators. In general, however, this is not so and the reason why the ML method is so widely adopted lies in the good properties of MLEs. The ﬁrst can be called the ‘covariance’ property with respect to parameter transformations; in fact, referring to eq. (5.4) we have Proposition 5.13 If qˆ = (θˆ1 , . . . , θˆk ) is the MLE of q = (θ1 , . . . , θk ) and h a ˆ is the MLE of h(q). one-to-one mapping from to (, ⊂ Rk ), rˆ = h(q) The proof is immediate because the function h−1 : → exists and Max L(q) = Max L(h−1 (r)) ≡ Max Lr (r) q∈

r∈

r∈

(we note in passing that the explicit form of Lr is obtained by simply setting h−1 (r) in the original likelihood function L; the differential elements must not be included because we transform the parameters and not the variables). So, for instance, the fact that S2 is the MLE of σ 2 in a,normalmodel with known mean and unknown variance tells us that S = {n−1 (Xi − µ)2 } is the MLE of the standard deviation σ . A useful consequence of Proposition 5.13, moreover, is that some problems can be cast in a simpler form by an appropriate change of parameters; in these cases we can solve the simpler problem – thus ﬁnding the ML estimates rˆ = (ˆr1 , . . . , rˆk ) – and then determine q = (θ1 , . . . , θk ) by means of h−1 . A nice example of this is given in Ref. [19]

200 Mathematical statistics (Chapter 2, Example 2.22) where a bivariate normal model (see eqs (3.61a) and (3.61b)) is considered and the ML estimates of σ 2 = θ1 and ρ = θ2 are determined by introducing the new parameters r1 = −[2σ 2 (1 − ρ 2 )]−1 and r2 = ρ[σ 2 (1 − ρ 2 )]−1 . Then, the desired result σˆ 2 = (2n)−1 ρˆ = 2

x2i + yi2

i

xi y i /

i

x2i + yi2

(5.62)

i

is obtained with a noteworthy simpliﬁcation of the calculations. A ﬁnal remark on Proposition 5.13: some authors speak of ‘invariance’ property. This term, however, would imply that the MLEs remain unchanged; since, in fact, they do change according to the transformation law h, we think that the term ‘covariance’ should be preferred. Other properties concern the relation between MLEs, efﬁcient estimators and sufﬁcient statistics, stated by the following two results, respectively. Proposition 5.14

ˆ If a MVUE T(X) of θ exists, then T(X) = T(X).

For regular problems, in fact, if a MVUE of θ exists it satisﬁes eq. (5.41). This, together with the likelihood equation (5.61) yields the desired result. ˆ Proposition 5.15 If T(X) is a sufﬁcient statistic for θ and the MLE T(X) ˆ of θ exists and is unique, then T is a function of T. The proof is almost immediate: since T is sufﬁcient, Neyman’s factorization (5.52) holds and maximizing L is equivalent to maximizing g which, in turn, depends on T. Consequently, the MLE itself will be a function of T. Before turning to the asymptotic properties of MLEs – which will be the subject of the next section – we point out two facts and state without proof an interesting result worthy of mention. First, MLEs, although asymptotically unbiased (see the following section), are often biased. Second, the ML method can be used in cases more general than the one considered here, that is, independent drawings from a ﬁxed distribution. For instance, the example (taken from Ref. [19]) on parameter transformation and mentioned above is a case in which, in fact, independence does not hold. Finally, the following proposition [30] provides an interesting characterization of some probability distributions based on a ML estimate: Proposition 5.16 For n ≥ 3, let X be a sample from a continuous population with pdf of the form f (x − θ ) and let X(1) ≤ X(2) ≤ · · · ≤ X(n) be the corresponding order statistics. If Tˆ = i ai X(i) with ai ≥ 0 and a1 + · · · + an = 1 is the MLE of θ, then

Preliminary ideas and notions

201

(a) if a1 = · · · = an = 1/n, then f is a normal density; (b) if a1 + an = 1; a1 an > 0, then f is a uniform density; (c) if aj + aj+1 = 1; aj aj+1 > 0 with j ∈ {1, 2, . . . , n − 1}, then f is a Laplace density. In regard to point (c), we call a Laplace r.v. a continuous r.v. X whose pdf is

|x − α| 1 fX (x) = exp − 2β β

(5.63)

where x ∈ R and the two parameters are such that α ∈ R; β > 0. Its CF is ϕX (u) =

eiαu 1 + β 2 u2

(5.64)

while its mean and variance are, respectively E(X) = α Var(X) = 2β 2 5.5.1

(5.65)

Asymptotic properties of ML estimators

As a matter of fact, some important properties of MLEs are asymptotic in nature. Since their proofs, however, are generally rather lengthy, this section is limited to the statement of the main results. For details, the interested reader can refer to more specialized literature (see, for instance, [3, 17, 19, 26, 28]). Assuming, as it is often the case, that we are dealing with a regular problem (Section 5.4.1) and that the likelihood function Ln attains its maximum at an interior point of for all n (this, in other words, means that the MLE exists for all n), then: (1) Tˆ n → θ[P], that is, the MLE is (weakly) consistent; √ (2) the r.v. n(Tˆ n − θ ) converges in distribution to a normal r.v. with zero mean and variance 1/I(θ ) or, equivalently, the MLE Tˆ n is asymptotically normal with mean θ and variance given by the Cramer–Rao limit {nI(θ)}−1 (eq. (5.39)). Although we do not provide the proofs of the above statements, some comments are not out of place. First, it should be noted that result (1) can be strengthened and strong consistency (in the sense of a.s. convergence) can be proven (see, for instance, [1]). Second, we have noted in the preceding section that the ML method does not always lead to unbiased estimators;

202 Mathematical statistics they are, however, asymptotically unbiased because the bias – which, anyway, can generally be removed for ﬁnite values of n – tends to zero as n−1 when we let n → ∞. Besides the minor inconvenience of bias for ﬁnite n, a more important property is given in point (2) in regard to the variance of MLEs. In fact, if we introduce the notion of asymptotic efﬁciency e¯ T of an estimator T as eT = limn→∞ eT , then eTˆ = 1, meaning that MLEs are asymptotically efﬁcient. Now, this fact does not imply that MLEs are the only asymptotically normal and asymptotically efﬁcient estimators but it has been shown that, in general, MLEs have better efﬁciency properties for large values of n (Refs. [23, 24]). In regard to this last observation, we note in passing that (2) does not generally imply that Var(Tˆ n ) → {nI(θ )}−1 as n → ∞ (D-convergence does not imply convergence of the moments); however, for a large class of asymptotically normal estimators the variance can be expressed as Var(T) =

a2 (θ ) 1 + ··· + nI(θ ) n2

and the estimator with the minimum a2 (θ ) is to be preferred (second-order efﬁciency). Quite often it turns out that MLEs are such estimators. Another remark on result (2) is that cases where the asymptotic variance depends on the unknown parameter are rather common. An appropriate parameter transformation can ﬁx the problem by maintaining, at the same time, asymptotic normality. In fact, if h is a differentiable function with h = √ 0 then it can be shown that the variable n{h(Tˆ n ) − h(θ )} is asymptotically normal with zero mean and variance [h (θ )]2 /I(θ ). Enforcing the condition √ that this new variance equals a constant – say b2 – we get h (θ ) = b I(θ ) and therefore h(θ) = a + b

, I(θ ) dθ

(5.66)

where both constants a, b can be chosen so that h(θ ) is in simple form. Example 5.8(a) Consider a sample from a Poisson variable (eq. (4.1)) with unknown parameter λ = θ. Since ∂ 2 f /∂θ 2 = −x/θ 2 then I(θ) = E(x/θ 2 ) =

1 1 θ x e−θ = x 2 x! θ θ x

(5.67)

because the sum – being the mean of the parent r.v. X – equals θ. It follows from (5.67) that the Cramer–Rao limit is θ/n. On the other hand, the MLE of θ is obtained by taking the logarithm of eq. (5.53) and equating its derivative

Preliminary ideas and notions

203

to zero; the reader can easily check that the result is 1 Xi Tˆ n = M = n

(5.68)

i

whose variance is θ/n (eq. (5.10), taking into account that Var(X) = θ). So, as expected, the MLE is consistent and in this case it is also efﬁcient because (eq. (5.43)) eTˆ = 1. Moreover, from result (2) we know that √ ˆ n(Tn − θ ) is asymptotically normal with zero mean and a variance which depends on the parameter, that is, 1/I(θ ) = θ. Setting a = 0) and b = 1 in √ √ √ eq. (5.66) we get h(θ ) = 2 θ so that the new variable 2 n( Tˆ n − θ ) is asymptotically standard-normal, that is, with zero mean and unit ) variance. √ √ Alternatively, setting a = 0 and b = 1/2 we have that Yn = n( Tˆ n − θ ) is asymptotically normal with zero mean and Var(Yn ) = 1/4. Example 5.8(b) When the model is non-regular, asymptotic normality may not hold. As an example, in the uniform model f (x; θ ) = 1/θ for 0 ≤ x ≤ θ (and zero otherwise) the likelihood function is

L(x; θ ) =

⎧ ⎨1/θ n ,

x(n) ≡ max xi ≤ θ

⎩0,

otherwise

1≤i≤n

and T = X(n) – where X(n) is the nth order statistic – is a sufﬁcient statistic for θ. Also, the likelihood function is monotone decreasing for θ ≥ x(n) and therefore it attains its maximum at θ = x(n) where, however, there is a discontinuity. So, even if we can call T = X(n) the MLE of θ, this is not a solution of the likelihood equation (5.61) and we may not expect property (2) to hold. In fact, we already know from Section 5.3.1 that the extreme value of the sample X(n) is not asymptotically normal. The above results still hold in the case of several parameters. Explicitly, referring to the considerations at the end of Section 5.4.1, property (2) becomes √ ˆ (2 ) the r.v. n(T n − q) is asymptotically normal with zero mean and variance {I(q)}−1 or, in case we are estimating a scalar function τ (q) = τ (θ1 , . . . , θk ) of the unknown parameters: √ (2 ) the r.v. n{Tˆ n − τ (q)} is asymptotically normal with zero mean and variance dT I−1 d, where d(q) = (∂τ/∂θ1 , . . . , ∂τ/∂θk )T .

204 Mathematical statistics

5.6

Interval estimation

Within the framework of the statistical model (5.1), we have discussed in the preceding sections the subject of ‘point estimation’ which, in essence, consists in (a) ﬁnding a ‘good’ estimator T(X) of the unknown parameter θ and (b) using the data from the experiment – that is, the realization of the sample x – to calculate the numerical value t = T(x). Then, on the basis of a number of considerations on what is meant by ‘good’, we expect t to be a reliable estimate of θ (broadly speaking, we could call it our educated ‘best-guess’ on the true value of θ). The procedure above is well justiﬁed if the main question of the estimation problem is ‘what value should I use for θ?’. If, however, one is more interested in specifying a range of values within which he/she can conﬁdently expect θ to lie, then the method of ‘interval estimation’ provides a better way to tackle the problem. In perspective, moreover, one should consider that a point estimate is almost meaningless without a statement of its ‘reliability’. So, still keeping the model (5.1) as our starting point, we now wish to determine an interval which contains the true value of θ – though unknown – at a speciﬁed ‘conﬁdence level’ (CL for short) γ = 1−α (0 < γ < 1). This, in other words, means that we have to ﬁnd two statistics T1 , T2 , with T1 < T2 , such that Pθ {T1 (X) < θ < T2 (X)} = γ

(5.69a)

for all θ ∈ . In this case we call (T1 , T2 ) a γ -conﬁdence interval (often γ -CI) for θ and T1 , T2 , respectively, the lower and upper conﬁdence limits. Note that eq. (5.69a) deﬁnes a random interval which, on the one hand, depends on the sample X but, on the other hand, does not depend on θ (because both limits are statistics). By carrying out an experiment we obtain a realization of the sample x and, accordingly, the values t1 = T1 (x) and t2 = T2 (x) for the two statistics; the interval (t1 , t2 ) is then an estimate of the γ -CI. At this point, one could be tempted to say that θ belongs to (t1 , t2 ) with a probability γ . This statement, however, is wrong because (t1 , t2 ) is not a random interval and therefore the true value of θ either belongs to it or it does not. The correct interpretation must be given in terms of relative frequency of success: if the experiment is repeated many times – thus obtaining many estimates of (T1 , T2 ) – the resulting estimated intervals will contain the true value of θ in 100γ % of the cases. Conversely, in the long run we will be wrong in 100α% of the cases. This, in essence, is the meaning of the term ‘conﬁdence’ in this context. Now, before showing how to determine the conﬁdence limits, some additional remarks on eq. (5.69a) are in order: (i) If the population under study is discrete it may not be possible to meet condition (5.69a) exactly; in this case we call γ -CL the smallest interval

Preliminary ideas and notions

205

such that Pθ {T1 (X) < θ < T2 (X)} ≥ γ

(5.69b)

for all θ ∈ . (ii) The statistic Dγ (X) = T2 − T1 is the length of the CI. This quantity can be considered as a measure of precision of our estimate: given, say, two methods of interval estimation and a CL γ , the method leading to the smaller Dγ is to be preferred. Whichever the adopted method, however, it is reasonable to expect that there must be a relation between D and γ because – for a ﬁxed sample size n – a higher conﬁdence level (or, equivalently, a lower α) is paid at the price of a larger interval. In fact, choosing an unreasonably high value of γ generally leads to a CI which is too large to be of any practical use (and consequently to almost no information on θ). If we want a high CL and an interval of acceptable length we can, of course, increase the sample size. Since this operation is generally costly, it is evident that any procedure of interval estimation implicitly implies a compromise between conﬁdence level, interval length and sample size. (iii) Equation (5.69) deﬁnes a two-sided interval but in some applications one-sided intervals are required; these intervals have the form (−∞, T2 ) or (T1 , ∞). (iv) In case of several unknown parameters, the CI for an individual component, say θi , is still given by (5.69) and the same applies in case of a scalar function τ (q) of the unknown parameter(s). Clearly, θ is replaced by θi in the former case and by τ (q) in the latter. More specifically, a γ -conﬁdence region for the vector parameter q = (θ1 , . . . , θk ) is a random subset Cγ (X) ⊂ such that for all q ∈ we have Pq {q ∈ Cγ (X)} ≥ γ

(5.69c)

The general technique used to determine conﬁdence intervals is based on the search of a so-called pivot quantity. This is a r.v. of the form G(X; θ ) – that is, it depends on the sample and on the unknown parameter and therefore it is not a statistic – such that (1) its distribution fG does not depend on θ and (2) for every x the function G(x; θ ) is continuous and strictly monotone in θ. Then, given γ ∈ (0, 1) there are many ways in which we can choose g1 < g2 so that the relation Pθ {g1 < G(X; θ ) < g2 } =

g2

g1

fG (g) dg = γ

(5.70)

206 Mathematical statistics holds. If, for every x, we deﬁne T1 (x) and T2 (x) – with T1 < T2 – as the solutions (with respect to θ) of the equations G(x; θ ) = g1 and G(x; θ ) = g2 , respectively, eq. (5.70) is equivalent to eq. (5.69). Note that T1 , T2 are welldeﬁned because they are obtained by means of the inverse (with respect to θ) function G−1 , which, in turn, is well-deﬁned by virtue of condition (2). So, if G is monotonically increasing then T1 (x) = G−1 (x; g1 ) and T2 (x) = G−1 (x; g2 ) while, on the other hand, T1 (x) = G−1 (x; g2 ) and T2 (x) = G−1 (x; g1 ) if G is monotonically decreasing. The question at this point is how to construct a pivot quantity. A number of useful results given in Appendix C will be of help in this task (see also the following examples) but here we outline a general procedure. Suppose we are dealing with an absolutely continuous model; it can be shown that if the parent r.v. X has a PDF FX (x; θ ) which is continuous and strictly monotone in θ then G(X; θ ) = −

n

ln F(Xi ; θ )

(5.71)

i=1

is a pivot quantity for the interval estimation of θ. The proof, which we only outline here, is based on the fact if X has a continuous and monotonically increasing PDF F(x) then the chain of equalities FY (y) = P(Y ≤ y) = P{F(X) ≤ y} = P{X ≤ F −1 (y)} = F[F −1 (y)] = y shows that the r.v. Y ≡ F(X) has a uniform distribution on the interval (0, 1). Consequently, each r.v. F(Xi ; θ ) in (5.71) is uniformly distributed on (0, 1), − ln F(Xi ; θ ) has a (1, 1) distribution and G(X; θ ) has a (1, n) pdf, that is, fG (g) =

g n−1 e−g (n)

(5.72)

which does not depend on θ. Since G(X; θ ) is evidently continuous and monotone in θ, it follows that it is a pivot quantity. So, by taking (5.72) into account and choosing g1 , g2 such thateq. (5.70) holds, the solutions of the equations − ln F(xi ; θ ) = g1 and − ln F(xi ; θ ) = g2 give the desired conﬁdence interval. This last step, in practice, is often the most difﬁcult part. Before giving some examples, we mention the following useful result (whose proof is immediate): Proposition 5.17 If (T1 , T2 ) is a γ -CI for θ and h is a strictly monotone function, then h(T1 ) and h(T2 ) are the limits of the γ -CI for h(θ ). The interval is (h(T1 ), h(T2 )) if h is monotonically increasing and (h(T2 ), h(T1 )) if h is monotonically decreasing.

Preliminary ideas and notions

207

Example 5.9(a) Let X be a sample from a normal population with unknown mean µ = θ and known variance. In this case only a small effort is required √ to see that the r.v. G = n(M − θ )/σ is a pivot quantity (condition (1) above follows from the fact that G ≈ N(0, 1) – see Section 5.3.1, Proposition 5.1(b) – and therefore its pdf does not depend on θ). Consequently, our γ -CI has the form

g2 σ g1 σ (T1 , T2 ) = M − √ , M − √ (5.73) n n any two numbers such that g1 < g2 and (g2 ) − (g1 ) = γ where g1 , g2 are √ x 2 (where (x) = ( 2π )−1 −∞ e−t /2 dt is the PDF of a standard normal r.v.). The shortest interval can be obtained by minimizing the function σ Dγ (g1 , g2 ) = √ (g2 − g1 ) n

(5.74)

under the constraint (g2 ) − (g1 ) = γ (we note in passing that this is a rather rare case where Dγ does not depend on X). Using the well-known method of Lagrange undeterminate multipliers and taking into account that the standard normal pdf is an even function we get g1 = −g2 . Then, since (−x) = 1 − (x) it follows that (g1 ) = (1 − γ )/2 = 1 − (g2 ) and (g2 ) = (1 + γ )/2. By calling c(1+γ )/2 the (1 + γ )/2-quantile of the standard normal distribution, that is, c(1+γ )/2 = −1 [(1 + γ )/2] (this, in other words, is that particular value of g2 that minimizes the interval length) the desired γ -CI for the mean is

σ σ (T1 , T2 ) = M − c(1+γ )/2 √ , M + c(1+γ )/2 √ (5.75a) n n where the values of c(1+γ )/2 can be found in statistical tables. The interval length is in this case σ Dγ = 2c(1+γ )/2 √ n

(5.75b)

So, for instance, if γ = 0.95 then (1 + γ )/2 = 0.975 and we ﬁnd c0.975 = 1.960 while at a higher conﬁdence level, say γ = 0.99, we get (1 + γ )/2 = 0.995 and c0.995 = 2.576. As noted in point (ii) eq. (5.75b) shows that a higher CL, for a given sample size n, is paid at the price of a longer interval; for a given conﬁdence level, on the other hand, the interval length can only be reduced by increasing n. Suppose now that we had used the median Z instead of M. We have pointed out at the end of Section 5.3.1 normal with , that Z is asymptotically , mean µ and standard deviation σ π/2n, that is, r = π/2 times the standard deviation of M. If, just for the sake of the argument, we suppose that the

208 Mathematical statistics error of the approximation can be neglected (in other words, we pretend√ that the distribution of Z is exactly normal) we get the γ -CI (Z±c(1+γ )/2 rσ/2 n), which is longer than (5.75) although the risk of error is the same. Example 5.9(b) Consider now the (more frequent) case in which the vari2 (eq. (5.31)) is an unbiased estimator of σ 2 we ance is not known. Since S¯√ may think of using G = n(M − θ )/S¯ as a pivot quantity. In this case, however, it can be shown that G ≈ St(n − 1) and therefore the quantiles of the Student distribution (with n − 1 degrees of freedom) will have to be used in specifying our conﬁdence interval for the mean. The symmetry of the distribution suggests that we can parallel the considerations above on g1 , g2 and arrive at the CI S¯ S¯ (T1 , T2 ) = M − t(1+γ )/2;n−1 √ , M + t(1+γ )/2;n−1 √ (5.76) n n where, denoting by S(n−1) the Student PDF with n−1 degrees of freedom, we −1 [(1 + γ )/2]. The values of these quantiles are also have t(1+γ )/2;n−1 = S(n−1) easily found on statistical tables for ν (the number of degrees of freedom) up to 40–50. Tables for higher values of ν are not given because St(ν) → N(0, 1) as ν → ∞ and the normal approximation is already rather good for ν ≥ 30. ¯ and therefore the interNote that now Dγ depends on the sample (through S) val length is a r.v. which can only be determined after we have carried out our experiment. Nonetheless, also in this case we expect the considerations of point (ii) to hold. As a numerical example of cases (a) and (b) suppose that we test 20 similar products and obtain an average weight of M = 100.2 g. If we know that the population standard deviation is, say, σ = 4 g, the 95%-CI for M is (eq. (5.75))

4 4 100.2 − 1.96 √ , 100.2 + 1.96 √ 20 20

= (98.45, 101.95)

If, on the other hand, we make no assumptions on the variance and calculate it from the data obtaining, say, s¯ = 3.80 g, we use eq. (5.76) to get

3.8 3.8 100.2 − 2.093 √ , 100.2 + 2.093 √ 20 20

= (98.42, 101.98)

because for γ = 0.95, (1 + γ )/2 = 0.975 and we ﬁnd from the tables (for ν = 19) the quantile t0.975;19 = 2.093. Note that the second interval is larger than the ﬁrst even if the estimated standard deviation is smaller than the true σ . This situation may occur in practice because in the second

Preliminary ideas and notions

209

case the uncertainty on the standard deviation also plays a part. Moreover, if we carried out another experiment on other 20 items giving, by chance, M = 100.2, the ﬁrst interval would not change while the second will because ¯ of the new estimate of S. A further consideration on example (a) is that eq. (5.75b) gives us the possibility to determine the minimum sample size needed to achieve a speciﬁed ‘precision’ of our estimate at a given CL. In fact, if the ‘precision’ is measured by Dγ , there may be cases in which we do not want our CI to exceed √a given length L. This condition is expressed by the relation 2c(1+γ )/2 σ/ n ≤ L which can be solved for n to give n≥

2c(1+γ )/2 σ L

2 (5.77)

Example 5.10(a) Suppose that we are still dealing with a normal model; now, however, we know the mean µ and the variance is unknown. Setting θ = σ , this means that we are looking for a CI for the function τ (θ) = θ 2 . It is not difﬁcult to see that G(X; θ ) =

n 1 (Xi − µ)2 θ2

(5.78)

i=1

is a pivot quantity. Now, since (Xi − µ)/θ ≈ N(0, 1) it is known (Appendix C) that (Xi − µ)2 /θ 2 ≈ χ 2 (1) from which it follows that G(X; θ ) ≈ χ 2 (n) by the reproducibility property of the χ 2 distribution. Solving the equations G(x; θ ) = g1 and G(x; θ ) = g2 we get a CI of the form (T1 (X), T2 (X)) =

g2−1

(Xi − µ)

2

, g1−1

i

(Xi − µ)

2

(5.79)

i

where – denoting by Kn (x) the PDF of the distribution χ 2 (n) – g1 , g2 must satisfy the condition Kn (g2 ) − Kn (g1 ) = γ . A common choice is to select a so-called ‘central’ interval, that is, to choose g1 , g2 as the (1 ∓ γ )/2 quantiles of χ 2 (n), respectively. This gives 2 g1 = Kn−1 [((1 − γ )/2] = χ(1−γ )/2;n −1 2 g2 = Kn [((1 + γ )/2] = χ(1+γ )/2;n

(5.80)

so that the CI (5.79) is explicitly written as (T1 , T2 ) =

nS2

nS2

, 2 2 χ(1+γ )/2;n χ(1−γ )/2;n

(5.81)

210 Mathematical statistics and the values of the quantiles can be found on statistical tables. So, for instance, if we are looking for a 95%-CI and n = 20, then (1−γ )/2 = 0.025 2 and (1 + γ )/2 = 0.975. Since on tables of χ 2 quantiles we ﬁnd χ0.025;20 = 2 2 2 9.59 and χ0.975;20 = 34.17, our interval is (0.59S , 2.09S ). Two remarks on this example: ﬁrst, √ √ it is a direct consequence of Proposition 5.17 that the interval ( T1 , T2 ) – where T1 , T2 are as in (5.81) – is a γ -CI for the standard deviation σ . Second, using Lagrange’s method one can determine that the estimated interval (5.81) is not optimal, that is, is not the shortest one. A quantitative evaluation, however, is not immediate and requires a numerical solution. For a 95%-CI it can be shown that the shortest interval involves two quantities α1 , α2 such that α1 +α2 = 1−γ and the corresponding quantiles are 9.96 and 35.23 (instead of 9.59 and 34.17). Example 5.10(b) If, as it often happens, also the mean of the population is not known, a pivot quantity is given by (5.78) by simply substituting M in place of µ, that is, G(X; θ ) = (n − 1)S¯ 2 /θ 2 = (n − 1)S¯ 2 /τ (where, as above, τ (θ) = θ 2 ). In this case G(X; θ ) ≈ χ 2 (n − 1) and we get the CI for the variance n−1 n−1 2 2 ¯ ¯ (T1 , T2 ) = (5.82) S , 2 S 2 χ(1+γ χ(1−γ )/2;n−1 )/2;n−1 so that, for instance, for n = 20 and γ = 0.95 we ﬁnd in tables the two 2 2 2 2 quantiles χ(1+γ )/2;n−1 = χ0.975;19 = 32.85 and χ(1−γ )/2;n = χ0.025;19 = 8.907. As above, the central CI (5.81) is not the shortest interval but it is the most frequently used in practice. If, at this point we also want a CI for the mean, we proceed exactly as in Example 5.9(b) thus obtaining the interval (5.76) which – owing to the symmetry of the Student distribution – is the ¯ M + a2 S). ¯ shortest among all intervals of the form (M − a1 S, Example 5.11(a) From the preceding examples it appears that the determination of CIs for the (unknown) mean of a normal population involves (i) standardized normal quantiles if the variance is known or (ii) Student quantiles – with the appropriate number of degrees of freedom – if the variance is not known. Provided that collective independence of the r.v.s involved in the estimation problem applies, this is a general fact. Suppose in fact, that we want to ﬁnd a CI for the difference µ1 − µ2 where µ1 = θ1 , µ2 = θ2 are the means of two normal populations with variances σ12 , σ22 , respectively. Also, let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Ym ) be the samples taken from the two populations and M1 , M2 the two sample means. If the variances are known then we can exploit the fact that G=

M1 − M2 − (θ1 − θ2 ) ≈ N(0, 1) ) n−1 σ12 + m−1 σ22

(5.83)

Preliminary ideas and notions

211

and therefore G is a pivot quantity. Proceeding exactly as in Example 5.9(a) we obtain the CI :

⎛ ⎝M1 − M2 ± c(1+γ )/2

⎞ σ12 σ22 ⎠ + n m

(5.84)

If the variances are not known we use the estimators S¯ 12 , S¯ 22 (or S12 , S22 ) instead of the population variances. Using these estimators, it is convenient to introduce the ‘pooled’ variance Sp2 =

nS2 + mS22 (n − 1)S¯ 12 + (m − 1)S¯ 22 = 1 n+m−2 n+m−2

(5.85)

because it can be shown (Appendix C) that the r.v. G=

M1 − M2 − (θ1 − θ2 ) , Sp n−1 + m−1

(5.86)

is distributed as a Student variable with n + m − 2 degrees of freedom. This is our pivot quantity for the case at hand and we can parallel Example 5.9(b) to get the CI

, M1 − M2 ± t(1+γ )/2;n+m−2 Sp n−1 + m−1 9 m+n = M1 − M2 ± t(1+γ )/2;n+m−2 nS12 + mS22 mn(m + n − 2) (5.87)

where the second expression has been written in terms of the sample variances S12 , S22 . Example 5.11(b) As above, let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Ym ) be independent samples from normal populations with unknown variances σ12 = θ12 , σ22 = θ22 , respectively. Now we wish to determine a CI for the ratio τ (θ1 , θ2 ) = θ12 /θ22 . The pivot quantity for this problem is obtained by noting that (Appendix C) Z1 = (n − 1)S¯ 12 /σ12 ≈ χ 2 (n − 1) and Z2 = (m − 1)S¯ 22 /σ22 ≈ χ 2 (m − 1) so that the r.v. S¯ 2 /θ 2 1 G = 12 12 = ¯S /θ τ 2 2

S¯ 12 S¯ 2 2

(5.88)

212 Mathematical statistics has a Fisher distribution with n − 1 and m − 1 degrees of freedom. Solving (5.88) for τ we get an interval of the form

S¯ 2 S¯ 2 g2−1 12 , g1−1 12 S¯ 2 S¯ 2

(5.89a)

so that denoting by F(1−γ )/2;n−1,m−1 and F(1+γ )/2;n−1,m−1 , respectively, the (1 − γ )/2 and (1 + γ )/2 quantiles of the distribution Fsh(n − 1, m − 1) the desired CI for the variance ratio is

S¯ 12 /S¯ 22

S¯ 12 /S¯ 22

, F(1+γ )/2;n−1,m−1 F(1−γ )/2;n−1,m−1 S¯ 12 S¯ 12 /S¯ 22 = , F(1+γ )/2;m−1,n−1 2 F(1+γ )/2;n−1,m−1 S¯ 2

(5.89b)

where in the second expression we took into account the property F(1−γ )/2;n−1,m−1 = {F(1+γ )/2;m−1,n−1 }−1 . So, for instance, if γ = 0.90, n = 20 and m = 15 we ﬁnd F0.95;19,14 = 2.40 and F0.95;14,19 = 2.26 and our interval is (0.417S¯ 12 /S¯ 22 , 2.26/S¯ 12 /S¯ 22 ). Example 5.11(c) As an example of a non-normal model, consider a sample taken from an exponential population with unknown mean (i.e. the statistical model is expressed in terms of the pdfs f (x; θ ) = θ −1 e−x/θ ). Now, since C) 2Xi /θ ≈ Exp(2) = χ 2 (2) and Xi ≈ Exp(θ) it follows that (Appendix −1 2 therefore G = 2θ i Xi ≈ χ (2n). It is left to the reader to ﬁll in the easy details and arrive at the central CI

2

Xi

2

Xi

, 2 2 χ(1+γ )/2;2n χ(1−γ )/2;2n

=

2nM

2nM

, 2 2 χ(1+γ )/2;2n χ(1−γ )/2;2n

(5.90)

2 As a numerical example, let γ = 0.90 and n = 10. We ﬁnd χ(1+γ )/2;2n =

2 2 2 = 31.41 and χ(1−γ χ0.95;20 )/2;2n = χ0.05;20 = 10.85; consequently (0.64M, 1.84M).

At this point a remark on notation is in order: whenever we have spoken of quantiles we meant lower quantiles. Some statistical tables report lower quantiles, but some other tables do not. In other words, if FG is the PDF under consideration (Gaussian, Student, χ 2 , Fisher, or else, depending on

Preliminary ideas and notions

213

the problem) and fG its density, our convention so far is that g1 FG (g1 ) =

fG (g) dg = (1 − γ )/2 = α/2 −∞

(5.91)

g2 FG (g2 ) =

fG (g) dg = (1 + γ )/2 = 1 − α/2 −∞

(we recall that γ = 1 − α by deﬁnition) so that the area under the pdf to the left of g1 equals α/2 and we can say, equivalently, that g1 is the α/2-lower quantile or, as we did, the (1 − γ )/2-lower quantile. Similarly, g2 is the (1 − α/2)-lower quantile or, equivalently, the (1 + γ )/2-lower quantile. In fact, for instance, one often ﬁnds – e.g. see [25] – the interval (5.81) written 2 2 as (nS2 /χ1−α/2;n , nS2 /χα/2;n ). From the ﬁrst of eq. (5.91), however, it follows that the area to the right ∞ of g1 is 1 − α/2, that is, P(G > g1 ) = g1 fG dg = 1 − α/2. Since the value of the area to the right of a given point is used to deﬁne the so-called ‘upper quantile’ of a distribution, the other convention sees g1 is the upper (1 − α/2)-upper quantile. By the same token, g2 is the upper α/2-upper quantile. Obviously, nothing changes for the degrees of freedom. So, for instance, one can ﬁnd eq. (5.81) written in terms of upper quantiles as (nS2 /χα/2;n , nS2 /χ1−α/2;n ) and now, if we look for a 95%-CI with, say, n = 20, we ﬁnd (see, for instance, Table 4 on [4] or Table C in Appendix II of 2 2 2 2 [7]) χα/2;n = χ0.025;20 = 34.17 and χ1−α/2;n = χ0.975;20 = 9.59. Obviously, the resulting interval is the same as above. In the following, in order to avoid confusion, we will explicitly state which type of quantile we are using; it must be the analyst’s care to check the tables at his/her disposal. Besides this observation on symbolism, it may also be worth spending a few words on some other interesting aspects of interval estimation. We start with the vector parameter case, which was brieﬂy mentioned in remark (iv) at the beginning of this section. The general technique used to construct conﬁdence regions is based on the fact that eq. (5.69c) is equivalent to Pq {X ∈ H(q)} ≥ γ

(5.92)

where, for every q ∈ , the set H(q) is the subset of the sample space containing all those realizations x (i.e. all those values taken on by X) such that the conﬁdence region constructed with these x will include q. So, the desired conﬁdence region is found by determining the sets H(q) satisfying inequality (5.92). Since, for a given CL, the sets H(q) can be chosen in many ways, the conﬁdence region thus constructed is not unique and the problem remains of ﬁnding a ‘minimal’ conﬁdence region. In practice, one generally ﬁnds the

214 Mathematical statistics sets H(q) with the help of some vector statistic T(X) with known distribution. As an example, we can reconsider Example 5.10(b) – normal model with unknown mean and variance – where we determined separate CIs for the mean and the variance. If, however, one considers the two-dimensional vector parameter q = (µ, σ 2 ) = (θ1 , τ ), it is wrong to deduce that the rectangle delimited by the intervals (5.76) and (5.82) is a γ -conﬁdence region for q. This is because the pivot quantities used to construct the CIs are related. Since it can be shown [19] that for a normal population the components of the two-dimensional statistic T = (M, S2 ) are independent, we can use the results (5.76) and (5.82) to obtain the set 3 , 4 H(q) = x : n/τ |m − θ1 | < a; b < ns2 /τ < b

(5.93)

2 2 where a = t(1+γ1 )/2;n−1 , b = χ(1−γ and b = χ(1+γ . Moreover, 2 )/2;n−1 2 )/2;n−1

the quantities γ1 , γ2 – owing to the independence of M and S2 – must satisfy the condition γ1 γ2 = γ in order to have a γ -conﬁdence region. Solving the inequalities which deﬁne H(q) we ﬁnd τ > n(m − θ1 )2 /a and ns2 /b < τ < ns2 /b . In the (θ1 , τ )-plane, therefore, the conﬁdence region is the part of the plane bounded by the parabola τ = n(m − θ1 )2 /a and the two straight lines τ = ns2 /b and τ = ns2 /b . Returning now to the one-dimensional case, a second consideration is the answer to the question: given a point estimator T(X) (of θ) with known distribution FT (t; θ ), can we construct a CI for θ? Intuitively, the answer is yes and, in fact, it is so. Let us assume that FT (t; θ ) is continuous and monotone in θ. Then, for every value of θ ∈ it is possible to deﬁne two numbers t1 , t2 (t1 < t2 ) such that Pθ {t1 < T(X) < t2 } = FT (t2 ; θ ) − FT (t1 ; θ ) = γ

(5.94)

Although they are not random quantities (because they are two realizations of T(X)) , t1 , t2 will be different for different values of θ; consequently, we can write t1 (θ ), t2 (θ ) and note that these two functions will generally be monotonically increasing in θ (if t is any sort of reasonable estimate of θ, it should increase as θ increases). Moreover, in order to uniquely deﬁne t1 , t2 one generally seeks a central interval by choosing them so that FT (t1 ; θ ) = (1 − γ )/2 FT (t2 ; θ ) = (1 + γ )/2

(5.95)

In the (θ, t)-plane we will therefore be able to identify a region bounded by the two functions t1 (θ ), t2 (θ ). This region, by construction, is such that eq. (5.94) holds for any ﬁxed value of θ ∈ ; but the important point is that for any ﬁxed value of t it deﬁnes two values θ1 (t), θ2 (t) – that is, the intersection of the horizontal line t with the curves t1 (θ ), t2 (θ ) – such that the interval (θ1 , θ2 ), in the long run, will bracket θ in γ % of the cases. This is

Preliminary ideas and notions

215

precisely the notion of conﬁdence interval for θ and therefore (T1 (X), T2 (X)), where Ti (X) = θi (T(X)) for i = 1, 2, is the desired γ -CI. So, under the assumptions above, we can in practice proceed as follows: given T(X) we obtain the realization x and consequently the estimate t = T(x); then, solving for θ the equations FT (t; θ ) = (1−γ )/2 and FT (t; θ ) = (1+γ )/2 we determine the extremes θ1 and θ2 of the γ -interval. By so doing, in the long run, we will be wrong (1 − γ )% of the times. We close this section with a ﬁnal observation on the examples above where, as the reader has probably noticed, we often assume a normal population as the starting statistical model. Although, clearly, the assumption of normality is not always justiﬁed in practice, we just point out two facts in its favour: (a) it has been shown that moderate and, sometimes, even signiﬁcant departures from normality lead to acceptable results in many cases and (b) if we suspect serious departures from normality, there is always the possibility of trying a transformation of the parent r.v. X (see, for instance, √ Ref. [2]) because log(X), X or some other function of it are often more nearly normal. Nonetheless, it goes without saying that in practical cases it is always advisable to check the basic assumption itself by carrying out a preliminary normality test on the data (this aspect is delayed to Chapter 6). 5.6.1

Asymptotic conﬁdence intervals

Consider√a point estimator Tn (X) of the unknown parameter θ such that the r.v. n(Tn − θ ) is asymptotically normal with zero mean and variance 2 2 σ √ (θ). If σ (θ) is a continuous function then it can be shown [19] that n(Tn − θ)/σ (Tn ) → N(0, 1) [D] as n → ∞. Consequently, for all θ we have √

n|Tn − θ| < c → (c) − (−c) = 2(c) − 1 = γ (5.96a) Pθ σ (Tn ) where c ≡ c(1+γ )/2 is the (1 + γ )/2-quantile of the standard normal distribution introduced in Example 5.9(a) and σ (Tn ) is the standard deviation of Tn . Since the relation above can be rewritten as Pθ

σ (Tn ) σ (Tn ) Tn − c(1+γ )/2 √ < θ < Tn + c(1+γ )/2 √ n n

→γ

(5.96b)

√ it follows that (Tn ± c1+γ /2 σ (Tn )/ n) is an asymptotic γ -CI for θ, where it is evident that the smaller is σ (Tn ) the shorter is the interval. As a consequence, asymptotically efﬁcient estimators will give the asymptotically shortest interval. If we recall from Section 5.5.1 that for regular models maximumlikelihood estimators are (i) asymptotically normal and (ii) asymptotically efﬁcient with variance 1/nI(θ ) = 1/In (θ ) – that is, the Cramer–Rao

216 Mathematical statistics limit – then the interval

c(1+γ )/2 Tˆ n ± √ nI(θ )

(5.97a)

(where Tˆ n is the ML estimator of θ) is the asymptotically shortest γ -CI for θ. Then, in order to ‘stablize’ the variance – that is, make it independent on θ – one may proceed as in Section 5.5.1 (eq. (5.66) and Example 5.8(a)) to obtain the conﬁdence interval for h(θ )

√ h(Tˆ n ) ± c(1+γ )/2 b/ n

(5.97b)

where, for simplicity, we chose a = 0 in eq. (5.66). If h is a monotone function we can then solve the resulting inequalities for θ to get the desired asymptotic γ -CI for the parameter θ. Owing to their nature, asymptotic CIs are exact only in the limit of n → ∞ but in common practice they are often used as approximate conﬁdence intervals when the sample is large – with the obvious understanding that the larger is the sample, the better is the approximation. As it should be expected, however, the notion of ‘large’ sample depends on the problem at hand because the rate of convergence to the normal distribution is not the same for all estimators. Nonetheless, it is a widely adopted rule of thumb that n > 30 can be considered a large sample when estimating conﬁdence intervals for means while n > 100 is the ‘dividing line’ between small and large samples when estimating conﬁdence intervals for variances. Example 5.12(a) In Example 5.8(a), we determined that the sample mean M is the ML estimator of the parameter θ of a Poisson model. Also, we found I(θ) √ = 1/θ noted that – choosing a = 0 and b = 1/2 in eq. (5.66) – the √ and √ r.v. n( M − θ ) is asymptotically normal √ with zero mean √ and variance 1/4. Then, it follows from eq. (5.97b) that ( M ± c(1+γ )/2 /2 n) is, for large √ samples, an approximate γ -CI for θ; consequently √ √ 2 √ √ 2 M − c(1+γ )/2 /2 n , M + c(1+γ )/2 /2 n

(5.98)

is the approximate γ -CI for θ. Example 5.12(b) For a sequence of n Bernoulli trials we have seen in Example 5.7(a) that the ML estimate of the parameter θ = p is the observed frequency of success x/n (which coincides with the sample mean M if 1 counts as a success and 0 counts as a failure). It is left to the reader to

Preliminary ideas and notions

217

show that I(θ) =

1 θ (1 − θ )

(5.99)

and therefore the approximate CI for θ is M±

c(1+γ )/2 , θ (1 − θ ) √ n

(5.100)

The stabilizing transformation can be obtained from eq. (5.66) which, setting a = 0 and b = 1/2, yields h(θ) =

1 2

√ dθ = arcsin( θ ) √ θ (1 − θ )

(5.101)

√ √ √ so that (arcsin( M) ± c(1+γ )/2 /2 n) is the approximate CI for arcsin θ.

5.7

A few notes on other types of statistical intervals

The somewhat detailed discussion of the preceding sections on conﬁdence intervals should not lead one to think that they are the only statistical intervals used in practice. Besides CIs, in fact, it is rather common in many applications to consider ‘tolerance intervals’ (TI) or ‘prediction intervals’ (PI), where the choice between the three types is dictated by the ﬁnal scope of the analysis. So, referring for the most part to Chapter 5 of [27], this section is simply meant to outline the main ideas behind these different concepts of statistical intervals. Before we do this, however, it is worth recalling that (a) the basic assumption is to draw a random sample from some population and (b) the statistical inferences are only valid for the population from which the sample was selected. In general, moreover, the assumption of normality is often made even if it may not be strictly met in practice. In this regard, the considerations at the end of Section 5.6 apply and in case of strong evidence of non-normality, one may always consider the possibility of using distribution-free methods (see, for instance, Ref. [13]). Tolerance intervals are needed when we are interested in an interval which will contain a certain percentage of the population. In this case, therefore, we will have two percentages: the percentage of population included in the interval and the conﬁdence level – often, as for CIs, 90, 95 or 99% – associated to the interval. This second percentage is usually included in the name and one speaks of 90%, 95% or 99%-TI, respectively. Assuming a sample from a normal population, tolerance intervals are ¯ and the values of cT,R – where the generally given in the form (M ± cT,R (n)S) subscript T is for ‘tolerance’ and R indicates the percentage of population contained in the interval – can be found in statistical tables for different

218 Mathematical statistics values of the sample size n. So, for instance, for n = 15 and a 95%-TI, we ﬁnd the values cT,90 = 2.48, cT,95 = 2.95 and cT,99 = 3.88. As the name itself implies, prediction intervals have to do with future observations. More speciﬁcally, a PI is needed when we are interested in an interval which will contain a speciﬁed number k of future observations from the population under study. So, for instance, given the population of daily ﬂights from, say, New York to Chicago, a pilot may not be interested in the average delay of these ﬂights, but in the delay of the next ﬂight in which he/she will be ﬂying. Similarly, a customer purchasing a small number of units of a given product is not interested in the long-run performance of the process from which his/her units are a sample, but in the quality of those particular units that he is buying. As for the other types of intervals, we associate to a PI a conﬁdence level but now the second deﬁning number is k, the number of future observations to be ¯ included in the interval. Again, the interval is given in the form (M±cP,k (n)S) where the subscript P is for ‘prediction’ and the values of cP,k can be found in statistical tables. As a numerical example, suppose that we have n = 10 observations from a normal population and we are interested in the values of k = 2 further randomly selected observations from that population. For n = 10, at a 95% CI we ﬁnd the value cP,2 = 2.79 so that our 95%-PI is ¯ where M and S¯ are the mean and (unbiased) standard deviation (M ± 2.79S), calculated on the basis of the ten observations at our disposal. An important difference between the types of intervals is that CIs become smaller and smaller as the sample size increases while it is not so for TIs and PIs. Finally, it is worth noting that there exist other types of prediction intervals such as, for instance, the PI to contain – at a given conﬁdence level – the mean of k future observations or the standard deviation of k future observations. For more detailed information the interested reader can refer to [13 and 14].

5.8

Summary and comments

The theory of Probability is an elegant and elaborate construction well worthy of study in its own right. Statistics, broadly speaking, is the other face of the coin because it provides the methods and techniques by which – on the basis of a limited number of observed data – we can make (inductive) inferences and/or draw conclusions on speciﬁc real-word problems where randomness is involved. In other words, one can safely say that Statistics ‘sees these problems from a different angle’, although it is evident that it must necessarily rely on Probability theory in order to be effective. The approach of Statistics is explained in Section 5.2, where the concept of statistical model is introduced together with the deﬁnitions of ‘sample’, ‘realization of the sample’ and some notes on the important aspect of data collection. With Section 5.3 we turn to more practical considerations by noting that one of the ﬁrst step in every analysis is to use the experimental data to

Preliminary ideas and notions

219

calculate the so-called ‘sample characteristics’ where, by analogy, each one of them is generally the counterpart of a well-deﬁned probabilistic quantity. In this light, therefore, one speaks of sample mean, sample variance, kth order (ordinary and central) sample moment, etc., and of their realizations which, in turn, may change from experiment to experiment because the realization of the sample, as a matter of fact, does change from experiment to experiment. Being random variables themselves, moreover, it makes sense to speak of mean, variance, etc. – and, more generally, of the probability distribution – of sample characteristics. All these aspects are discussed in Section 5.3 by implicitly assuming the sample size n as ﬁxed. This, however, is not the whole story because another important issue is considered in Section 5.3.1: the behaviour of sample characteristics as the sample size increases indeﬁnitely – that is, mathematically speaking, as n → ∞. In the limit, in fact, some important properties of both theoretical and practical interest show up: theoretical because an inﬁnite sample is an evident impossibility and consequently these asymptotic properties can never be realized in full, but practical because it can often be assumed that they are, to a certain extent, satisﬁed by large samples, thereby providing useful working approximations in many cases. Having introduced the concept of sample characteristic and, in particular, of statistic – that is, a sample characteristic containing no unknown quantities – both Sections 5.4 and 5.5 and all their subsections are dedicated to the subject of point estimation. In essence, the problem consists in estimating one or more unknown parameters of a supposedly known type of distribution by means of an appropriate statistic. The type of distribution provides the underlying statistical model while the observed data are used to calculate the ‘appropriate’ statistic which, we hope, will estimate the unknown parameter(s) within an acceptable degree of accuracy. Since this kind of problem is fundamental in almost all statistical applications, the ﬁrst step is to specify some criteria by which we may be able to decide whether a given statistic can qualify as a ‘good’ – or even, if and when possible, as the ‘best’ – estimator for the parameter under investigation. In this respect, in fact, it is not sufﬁcient to rely solely on analogy – that is, using the sample mean to estimate the mean, the sample variance for the variance, etc. – because it can be shown that this intuitive approach, although useful in some cases, may even be misleading in some other cases. Among the most important criteria to judge an estimator, Section 5.4 considers unbiasedness, asymptotic unbiasedness, efﬁciency and consistency. Then, in regard to efﬁciency, Section 5.4.1 deals with a fundamental result applying to the so-called regular problems: this is the Cramer–Rao inequality which, by establishing a lower limit for the variance of an estimator, can indicate the best estimator – when it exists – in terms of efﬁciency. In the process, the deﬁnition of Fisher’s information is given and all the concepts above are generalized to the case of a k-dimensional (vector) parameter and to a scalar function of a vector parameter.

220 Mathematical statistics Another desirable property of estimators is sufﬁciency. The deﬁnition is not self-evident and, often, is also of little practical use for the purpose of identifying sufﬁcient estimators. The required explanations are given in Section 5.4.2, where it is also shown that Neyman’s factorization theorem provides an easier way to assess sufﬁciency and that – Rao–Blacwell theorem – the so-called MVUE (minimum variance unbiased estimator), when it exists, is a function of a sufﬁcient statistic. The property of completeness, moreover, is introduced in order to state Lehmann–Scheffé theorem which, in turn, speciﬁes the general form of a MVUE as a function of a sufﬁcient and complete statistic and an unbiased estimator (for the unknown parameter under study). Finally, the way in which we can ﬁnd estimators with the above properties – or at least some of them – is explained in Section 5.5. Although not the only one, the most popular technique for this purpose is the so-called ML method. The name itself is self-explanatory and consists in maximizing the likelihood function (or, more often, its natural logarithm) with respect to the unknown parameter(s). The ‘method of moments’ and ‘Bayes’ method’ are also brieﬂy considered in Section 5.5 but it is noted that, in general, ML estimators have a number of desirable properties and here, probably, lies the reason for the method’s popularity. Particularly worthy of mention are the asymptotic properties of ML estimators considered in Section 5.5.1. The most appropriate solution to many problems is not in the form of a point estimate because the main concern is often a range of values within which we can conﬁdently hope to ﬁnd the true value of the unknown parameter. This is a so-called problem of interval estimation and is the subject of Section 5.6. So, by ﬁrst specifying a conﬁdence level γ , our goal is to determine two statistics T1 , T2 such that eq. (5.69) holds; these statistics, once we ﬁnd them, are the lower and upper limit of the CI, respectively. At this point we use the experimental data to calculate their realizations t1 , t2 and say that (t1 , t2 ) is the desired γ -CI. The general technique by which the task of ﬁnding T1 , T2 is accomplished is explained in Section 5.6 and the many worked-out examples show that conﬁdence intervals are always speciﬁed in terms of quantiles of an appropriate distribution where, on the one hand, the ‘appropriate’ distribution (frequently the Gaussian, the χ 2 or the Fisher distribution) depend on the parameter under study while, on the other hand, the quantiles to be used in actually calculating the interval depend on the conﬁdence level. In any case, however, it is pointed out that we cannot say that the true value θ of the parameter lies in the interval (t1 , t2 ) with probability θ. This is because (t1 , t2 ) is a ‘deterministic’ interval with nothing random in it and therefore θ either belongs to it or it does not. The correct statement is given in terms of the long-run interpretation of conﬁdence intervals: by repeating the estimation procedure many times – thus obtaining many conﬁdence intervals – θ will fall in these intervals in 100γ % of the cases. Also, another general fact is that the procedure of interval estimation must be based on a compromise

Preliminary ideas and notions

221

between sample size and conﬁdence level. For a given sample size, in fact, a higher conﬁdence level corresponds to a longer interval and therefore an unreasonably high value of γ will lead to an interval which may be too large to be of any practical use. The interval length, on the other hand, can be decreased by either choosing a lower conﬁdence level or by increasing the sample size, or both. Increasing the sample size, however, is generally costly and, in some cases, may not even be practicable. So, a correct balance of these quantities must be agreed upon at the planning stage and, clearly, it is the analyst’s responsibility – depending on the importance of the problem at hand – to suggest a viable solution. Finally, it is noted that cases in which ﬁnding a conﬁdence interval turns out to be a very difﬁcult task are not rare. For large samples, however, a practical solution is the use of asymptotic conﬁdence intervals and this is the subject of Section 5.6.1. In Section 5.7, moreover, we brieﬂy introduce the concepts of ‘tolerance intervals’ and ‘prediction intervals’ by also giving a number of speciﬁc references for the reader interested in more details on these further aspects of interval estimation.

References and further reading [1] Azzalini, A., ‘Inferenza Statistica: una Presentazione Basata sul Concetto di Verosimiglianza’, 2nd edn., Springer-Verlag Italia, Milano (2001). [2] Bartlett M.S., ‘The Use of Transformations’, Biometrics, 3, pp. 39–52 (1947). [3] Cramér, H., ‘Mathematical Methods of Statistics’, 19th edn., Princeton Univ. Press, Princeton (1999). [4] Crow, E.L., Davis, F.A., Maxﬁeld, M.W., ‘Statistics Manual’, Dover, New York (1960). [5] de Haan, L., ‘Sample extremes: an Elementary Introduction’, Stat. Neerlandica, 30, 161–172 (1976). [6] Di Crescenzo, A., Ricciardi, L.M., ‘Elementi di Statistica’, Liguori Editore, Napoli (2000). [7] Duncan, A.J., ‘Quality Control and Industrial Statistics’, 5th edn., Irwin, Homewood, Illinois (1986). [8] Edwards, A.W.F., ‘Likelihood’, The Johns Hopkins University Press, Baltimore (1992). [9] Fisher, R.A., ‘On the Mathematical Foundations of Theoretical Statistics’, PTRS, 222 (1921). [10] Galambos, J., ‘The Asymptotic Theory of Extreme Order Statistics’, 2nd edn., Krieger, Malabar (1987). [11] Gnedenko, B.V., ‘Sur la Distribution Limite du Terme Maximum d’une Série Aléatoire’, Ann. Math., 44, 423–453 (1943). [12] Green, J.R., Margerison, D., ‘Statistical Treatment of Experimental Data’, Elsevier, Amsterdam (1977). [13] Hahn, G.J., Meeker, W.Q., ‘Statistical Intervals: a Guide for Pratictioners’, Wiley, New York (1990). [14] Hahn, G.J., ‘Statistical Intervals for a Normal Population. Part I. Tables, Examples and Applications’, Journal of Quality Technology, July, 115–125

222 Mathematical statistics

[15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30]

(1970); ‘Part II. Formulas, Assumptions, some Derivations’, Journal of Quality Technology, October, 195–206 (1970). Huff, D., ‘How to Lie With Statistics’, W. W. Norton & Company, New York (1954). Keeping, E.S., ‘Introduction to Statistical Inference’, Dover, New York (1995). Klimov, G., ‘Probability Theory and Mathematical Statistics’ Mir Publishers, Mosow (1986). Kottegoda, N.T., Rosso, R., ‘Statistics, Probability and Reliability for Civil and Environmental Engineers’, McGraw-Hill, New York (1998). Ivchenko, G., Medvedev, Yu., ‘Mathematical Statistics’, Mir Publishers, Moscow (1990). Mandel, J., ‘The Statistical Analysis of Experimental Data’, Dover, New York, (1984). Mendenhall, W., Wackerly, D.D., Scheaffer, R.L., ‘Mathematical Statistics with Applications’, 4th end., PWS-KENT Publishing Company, Boston (1990). Nasri-Roudsari, D., Cramer, E., ‘On the Convergence Rates of Extreme Generalized Order Statistics’ www.math.uni-oldenburg.de/preprints/get/source/ Rates.pdf Pace, L., Salvan, A., ‘Teoria della Statistica’, CEDAM, Padova (1996) Rao, C.R., ‘Asymptotic Efﬁciency and Information’, Proc. 4th Berkeley Symp. Math. Stat. Prob. 1, 531–545 (1961). Rinne, H., ‘Taschenbuch der Statistik’, Verlag Harri Deutsch, Frankfurt am Main (2003). Serﬂing, R.J., ‘Approximation Theorems in Mathematical Statistics’, John Wiley & Sons, New York (1980). Wadsworth, H.M. (editor), ‘Handbook of Statistical Methods for Engineers and Scientists’, McGraw-Hill, New York (1990). Zacks, S., ‘The Theory of Statistical Inference’, John Wiley & Sons, New York (1971). Thisted, R.A., ‘Elements of Statistical Computing’, Chapman & Hall, London (1988). Buczolich Z., Székely G.J., Adv. Appl. Math., 10, 439–256 (1992).

6

6.1

The test of statistical hypotheses

Introduction

Broadly speaking, any assumption on the distribution of one or more random variables observed in an experiment is a statistical hypothesis. The hypothesis may be based on theoretical considerations, on the analysis of other (similar) experiments or it may just be an educated guess suggested by reasonableness or common sense, whatever these terms mean. In any case, it must be checked by actually performing the experiment and by devising some method which – in the light of the acquired data – gives us the possibility to decide whether to accept it or reject it. This, it should be clear from the outset, does not imply that our decision will be right because, as in any procedure of statistical inference, the best we can do (unless we examine the entire population) is to reduce the probability of being wrong to an acceptable level, where the term ‘acceptable’ generally depends on the problem at hand, the seriousness of the consequences of being wrong and, last but not least, the cost of the experiment. Consequently, we will not state our conclusions by saying ‘our hypothesis is true (false)’ but ‘the observed data are in favour (against) our hypothesis’, and we will continue our work behaving as if the hypothesis were true (false). The methods by means of which we make our decision are called statistical tests and are the subject of this chapter. We will ﬁrst illustrate the main ideas from a general point of view and then turn to typical classes of problems and speciﬁc examples.

6.2

General principles of hypotheses testing

Let us start with some deﬁnitions. The hypothesis to be tested, generally denoted by H0 , is called the null hypothesis and it is tested against an alternative hypothesis H1 . The two hypotheses are regarded as mutually exclusive and exhaustive. This is to say that if we accept H0 then we reject H1 and conversely, but it does not mean that – given H0 – the hypothesis H1 is the one and only alternative to H0 . As a matter of fact, it is often possible to conceive of several alternatives to H0 , say H1 , H1 and so on, but the

224 Mathematical statistics point is that once H0 and H1 have been formulated, the test leading to the acceptance/rejection of H0 necessarily leads to the rejection/acceptance of H1 . Clearly, it is the analyst’s responsibility to select the most appropriate pair of hypotheses for the problem at hand. So, given H0 and H1 , the experimental data form the evidence on the basis of which we decide to accept or reject H0 . Due to the intrinsic uncertainty of any statistical inference, our decision may be right or wrong; however, we may be wrong in two ways: (a) by rejecting H0 when in fact it is true; or (b) by accepting H0 when in fact it is false. The common terminology deﬁnes (a) a type I error (or rejecting error) and (b) a type II error (or acceptance error). Ideally, one would like both possibilities of error to be as small as possible, but since it turns out that, for a ﬁxed sample size n, it is generally not possible to decrease one type without increasing the other, some sort of compromising strategy must be adopted. We will come to this point shortly. In essence, any statistical test is a rule by which a realization x = (x1 , . . . , xn ) of the sample X = (X1 , . . . , Xn ) is used to make a decision about the assumption H0 . More speciﬁcally, this is done by dividing the sample space into two disjoint sets 0 , 1 – called the acceptance region and the rejection (or critical) region, respectively – such that 0 ∪ 1 = . As the names themselves imply, 0 contains all x which lead to the acceptance of H0 while 1 contains all x which lead to the rejection of the null hypothesis. In this light, the basic formulation of a statistical test is as follows: Let x be a realization of the sample X. If x ∈ 0 we accept the null hypothesis H0 ; if, on the other hand, x ∈ 1 we reject H0 (and therefore accept H1 ). Then, the two possibilities of error correspond to the cases: (a) x ∈ 1 when H0 is true and (b) x ∈ 0 when H0 is false. The selection of the acceptance and rejection regions is strictly related to two other aspects: the test chosen for a given null hypothesis and the ‘goodness’ of the test. In fact, since it is reasonable to expect that a given null hypothesis H0 can be tested by different methods and that each method will deﬁne its acceptance and rejection regions, the problem arises of which test to choose among all possible tests on H0 . The choice, we will see, depends also on the alternative hypothesis H1 but for the moment we assume both H0 and H1 as given. Now, an intuitive solution to this problem is, for a speciﬁed sample size n, to call ‘best’ the test which makes the possibility of error as small as possible and choose this one. This aspect, however, deserves further consideration because – keeping in mind that we do not know if H0 is true or not – the two types of error must be considered simultaneously. If, as it is customary, we denote by α and β the probabilities of committing a type I and type II error respectively, it turns out that we cannot simultaneously make them as small as we wish. This fact is evident if we examine

The test of statistical hypotheses

225

the two extreme cases. If we choose α = 0 we will never make a type I error and this, in turn, means that = 0 because we will always accept H0 regardless of the observed realization x. This is a correct decision if H0 is true (which we do not know); if, however, H0 is false, our choice of accepting it no matter what – since 1 = C 0 = ∅ – implies β = 1. Conversely, choosing β = 0 means that = 1 and 0 = ∅; therefore we will always reject H0 , a circumstance which prevents us from committing a type II error if H0 is false, but implies α = 1 if H0 is true. Between the two extremes there are many possible intermediate cases corresponding to different choices of 0 and 1 but it is a general fact that reducing α tends to increase β and viceversa. The usually adopted strategy to overcome this difﬁculty is due to Neyman and Pearson and is based on the consideration that in most cases one type of error has more serious consequences than the other. Consequently, we ﬁx a value for the probability of the worst error and, among all possible tests, we choose the one that minimizes the probability of the other error. Since the problem is often formulated in such a way that the type I error is the worst, the strategy consists in specifying a value for α and – if and when possible – choosing the test with the smallest value of β (or, equivalently, the maximum value of 1 − β) compatible with the prescribed value of α. This speciﬁed value of α – which, clearly, depends on practical considerations about the problem at hand – deﬁnes the signiﬁcance level of the test. Before turning to other general aspects of hypothesis testing, we open a short parenthesis on notation. Often one denotes the probabilities of type I and type II errors by P(H1 |H0 ) and P(H0 |H1 ), respectively. This symbolism does not mean that we are dealing with conditional probabilities in the strict sense, but it is just a convenient way of indicating – in the two cases – the accepted hypothesis (in the ﬁrst ‘slot’ within parenthesis) and the true hypothesis (in the second ‘slot’). Returning to the main discussion, an observation of practical nature is that the critical (rejection) region is frequently deﬁned by means of a so-called test function T(X), where T(X) is a statistic which must be appropriately chosen for the problem at hand. Having chosen a test statistic, the critical region will then be expressed in one of the following forms ⎧ ⎪ ⎨{x : T(x) ≥ c} 1 = {x : T(x) ≤ c} ⎪ ⎩ {x : |T(x)| ≥ c}

(6.1)

where c is a real number which depends on the signiﬁcance level α. This, in other words, means that for every α the set T = {t : t = T(x), x ∈ } of all possible values of T is divided into two subsets T0 , T1 , where T1 will include all those realizations t = T(x) which lead to the rejection of H0 .

226 Mathematical statistics A second comment worthy of mention is that, for a ﬁxed sample size n, some problems of hypothesis testing do not lend themselves easily to a solution. Things, however, often get better if we adopt an asymptotic approach by letting the sample size tend to inﬁnity. This is a frequently adopted strategy but it should be kept in mind that the ﬁnal results are then valid only for large samples. For moderate sample sizes, however, they can often be considered as useful working approximations. Having outlined the general philosophy of the statistical testing, it can be of help at this point to have an idea of some typical of types of hypotheses encountered in practice. The following is a short list: (1) Hypothesis on the form of distribution: In this case we make n independent observations of a r.v. X with unknown distribution FX (x) and use the acquired data x to check if the distribution of X is, as we assume, F(x). The null hypothesis is then written H0 : FX (x) = F(x). The function F(x), in turn, may be (a) completely deﬁned or (b) may belong to a certain class – for example, normal, Poisson, or else – the uncertainty being on one (or more) parameter(s) θ of the distribution. An example of this latter type can be H0 : FX (x) = N(µ, θ ), meaning that we want to test the hypothesis that X has a normal distribution with known mean µ and unknown variance θ. (2) Hypothesis of independence: In this case we have, for example, a twodimensional r.v. X = (X, Y) with unknown PDF FX (x, y) and we have reasons to believe that X and Y are independent. Then, the null hypothesis is symbolically expressed as H0 : FX (x, y) = FX (x)FY (y). (3) Hypothesis of homogeneity: We carry out a series of m independent experiments – each experiment consisting of n trials – obtaining the results (x1i , . . . , xni ), where i = 1, . . . , m. Our basic assumption in this case is that these data are homogeneous, that is, they are all observations of the same random variable. Then, since the null hypothesis is that the distribution law is the same for all the experiments, we symbolically express the problem as H0 : F1 (x) = F2 (x) = · · · = Fm (x), where we denoted by Fi (x) the (unknown) distribution of the ith experiment. Clearly, the types of hypothesis considered above do not cover all the possibilities because the list has been given mainly for illustrative purposes. Other speciﬁc cases will be examined in due time if and whenever needed in the course of future discussions.

6.3

Parametric hypotheses

If the hypothesis to be tested concerns one or more unknown parameters of a supposedly known type of probability distribution, one speaks of parametric hypotheses. The basic procedure is similar to what has been done in Chapter 5 – that is, we start from the statistical model (5.1) and, on the

The test of statistical hypotheses

227

basis of the acquired data, draw inferences on θ – but the details differ. In Chapter 5, in fact, we did not formulate any hypothesis whatsoever on θ and our main concern was simply to determine a reliable estimate of it, either in the form of a numerical value or a conﬁdence interval. Now we do formulate an hypothesis – the null hypothesis H0 – and the scope is to accept it or reject it depending on whether H0 is reasonably consistent with the observed data or not. This kind of approach is generally more convenient when, following the experiment, we must make a ‘yes or no’ decision and take action accordingly. For the moment we ignore the fact that the two problems – parametric hypothesis testing and conﬁdence interval estimation – are, in fact, related and delay the discussion of this aspect to Section 6.3.4. Denoting, as in Chapter 5, the parameter space by , the general form of the null and alternative hypotheses is H0 : θ ∈ 0 H1 : θ ∈ 1

(6.2)

where 0 , 1 are two subsets of such that 0 ∩ 1 = ∅ and 0 ∪ 1 = . More speciﬁcally, we call simple any hypothesis which speciﬁes the probability distribution completely, otherwise we speak of composite (or compound) hypothesis. So, for instance, H0 : θ = θ0 and H1 : θ = θ1 (where θ0 and θ1 are given numerical values) are simple hypotheses while H0 : θ ≥ θ0 , H1 : θ = θ0 or, say, H1 : θ < θ0 are composite hypotheses. Depending on the problem at hand, we may have any one of the three possibilities (i) both the null and alternative hypotheses are simple, (ii) one is simple and the other is composite and (iii) both hypotheses are composite. Before examining the various cases, we must return for a moment to the discussion of Section 6.2 on how to select a ‘good’ test, a choice which – we recall – requires a closer look at the two types of error. In case of parametric hypotheses, they generally depend on θ and can be written as α(θ) = Pθ (X ∈ 1 | θ ∈ 0 ) β(θ ) = Pθ (X ∈ 0 | θ ∈ 1 )

(6.3)

If we deﬁne the so-called power function W(θ ) as W(θ) =

Pθ (X ∈ 1 | θ ∈ 0 ) = α(θ) Pθ (X ∈ 1 | θ ∈ 1 ) = 1 − β(θ )

(6.4)

we recognize 1 − β as the probability of not making a type II error. Since an ideal test will result in W(θ ) = 0 if H0 is true (i.e. θ ∈ 0 ) and W(θ ) = 1 if H0 is false (i.e. θ ∈ 1 ), the function W can be used to compare different tests on a given pair of hypothesis H0 , H1 . In this light, in fact, we have the

228 Mathematical statistics following deﬁnitions: (i) we call size of a test the quantity α = sup W(θ )

(6.5)

θ∈0

(note that for parametric hypotheses the terms ‘size’ and ‘signiﬁcance level’ are interchangeable). (ii) given a test T on a pair of hypotheses H0 , H1 , let α be its size and β(θ ) its probability of a type II error. Then T is called the uniformly most powerful test if, for any other test T ∗ (on H0 , H1 ) of size α ∗ ≤ α, we have β ∗ (θ) ≥ β(θ ) for all θ ∈ 1 . This, in other words, means that the uniformly most powerful test T – denoting by W(θ ) its power function – satisﬁes the inequality W(θ) ≥ W ∗ (θ )

for all θ ∈ 1

(6.6)

Also, a desirable property for a test is unbiasedness. A test T is called unbiased if W(θ) ≥ α

for all θ ∈ 1

(6.7)

so that we have a higher probability of rejecting H0 when it is false than rejecting it when it is true. At this point, another word of caution is in order because mistakes and misunderstandings are rather frequent: the power function considers the probabilities of rejecting H0 when it is true and when it is false. This is, in essence, the main idea of hypothesis testing and nothing can be said about the probability of H0 being true or false. So, if H0 is accepted at, say, the 5% signiﬁcance level it does not mean that the probability of H0 being true is 95%. This distinction, as a matter of fact, is fundamental and should always be kept in mind when reporting the results. 6.3.1

Simple hypotheses: Neyman–Pearson’s lemma

A uniform more powerful test does not always exist because uniformity, that is, the condition ‘for all θ ∈ 1 ’, is rather strong and we may have cases in which two tests, say T1 , T2 , cannot be compared because W1 (θ ) < W2 (θ ) for some values of θ in 1 while W1 (θ ) > W2 (θ ) for some other values of θ in 1 . A most powerful test, however, always exists when we are dealing with a pair of simple hypothesis, that is, the case in which eq. (6.2) have

The test of statistical hypotheses

229

the form H0 : θ = θ0

(6.8)

H1 : θ = θ1

where θ0 , θ1 are two speciﬁc numerical values for the unknown parameter. Equation (6.8), in other words, imply that the parameter space consists of only two points – that is, = {θ0 , θ1 } – and that the distribution of the r.v. X is either F0 (x) = F(x; θ0 ) or F1 (x) = F(x; θ1 ), where F is a known type of PDF (normal, exponential, Poisson or else). Assuming that F0 and F1 are both absolutely continuous with densities f0 (x) and f1 (x), respectively (with f0 , f1 > 0), the following theorem – known as Neyman–Pearson’s lemma – holds. Proposition 6.1 (Neyman–Pearson’s lemma) Let (6.8) be the null and alternative hypotheses and x = (x1 , . . . , xn ) be a realization of the sample X = (X1 , . . . , Xn ). The most powerful test of size α is speciﬁed by the critical region 1 = {x : l(x) ≤ c}

(6.9)

where c (c ≥ 0) is such that Pθ0 [l(X) ≤ c] = α and l(X) is a statistic called the ‘likelihood-ratio’ and deﬁned as l(X) ≡

& f0 (Xi ) L(X; θ0 ) = &i L(X; θ1 ) i f1 (Xi )

(6.10)

In order to simplify the notation in the proof of the theorem let us call A the rejection region (6.9) and let B be the rejection region of another test of size α (on the hypotheses (6.8)). Then

L(x; θ0 ) dx =

A

L(x; θ0 ) dx = α B

because both tests have size α. Noting that both A and B can be expressed as the union of two disjoint sets by writing A = (A ∩ B) ∪ (A ∩ BC ) and B = (A ∩ B) ∪ (B ∩ AC ) respectively, the equality above implies

L(x; θ0 ) dx =

A∩BC

L(x; θ0 ) dx

(6.11)

B∩AC

By the deﬁnition of A, moreover, it follows that L(x; θ1 ) ≥ L(x; θ0 )/c for x ∈ A and, clearly, L(x; θ1 ) < L(x; θ0 )/c for x ∈ AC . Using these inequalities,

230 Mathematical statistics eq. (6.11) leads to the chain of relations

L(x; θ1 ) dx ≥

A∩BC

L(x; θ0 ) dx c

A∩BC

=

L(x; θ0 ) dx > c

L(x; θ1 ) dx B∩AC

B∩AC

which, in turn, are used to get

L(x; θ1 ) dx = A

L(x; θ1 ) dx + A∩B

>

L(x; θ1 ) dx

A∩BC

L(x; θ1 ) dx + A∩B

=

L(x; θ1 ) dx

B∩AC

L(x; θ1 ) dx

(6.12)

B

meaning that the probability 1 − β is higher for the test with rejection region A = 1 . In fact, since the quantity 1 − β (i.e. the probability of rejecting H0 when H0 is false or, equivalently, of accepting H1 when H1 is true) of a given test is obtained by integrating L(x; θ1 ) over its rejection region, eq. (6.12) proves the theorem because the test corresponding to B is any test of size α on the hypotheses (6.8). In addition, we can show that the test is always unbiased. In fact, in the rejection region A = 1 (eq. (6.9)) we have L(x; θ0 ) ≤ cL(x; θ1 ) which, if c ≤ 1, implies L(x; θ0 ) ≤ L(x; θ1 ) and therefore α=

L(x; θ1 ) dx = 1 − β = W(θ1 )

L(x; θ0 ) dx ≤ A

A

On the other hand, in the acceptance region AC we have L(x; θ0 ) > cL(x; θ1 ) and therefore L(x; θ0 ) > L(x; θ1 ) whenever c > 1. Consequently 1−α =

L(x; θ0 ) dx >

AC

L(x; θ1 ) dx = β = 1 − W(θ1 ) AC

thus showing that condition (6.7) holds in any case. Example 6.1(a) As an application of Neyman–Pearson’s lemma, consider a normal r.v. with known variance σ 2 . On the basis of the random sample X

The test of statistical hypotheses

231

and the observed data x we want to test the pair of simple hypotheses (6.8) on the unknown mean µ = θ where, for deﬁniteness, we assume θ1 > θ0 . We have n 1 [(xi − θ0 )2 − (xi − θ1 )2 ] l(x) = exp − 2 2σ i=1 nm n θ12 − θ02 − 2 (θ1 − θ0 ) = exp 2 2σ σ where m = xi is the realization of the sample mean calculated from the data. The inequality deﬁning the rejection region (6.9) holds if

m≥

(θ1 + θ0 ) σ 2 log c − 2 n(θ1 − θ0 )

(6.13a)

or, equivalently, if √ √ σ log c n(m − θ0 ) n(θ1 − θ0 ) ≥ −√ ≡ t(c) σ 2σ n(θ1 − θ0 )

(6.13b)

√ Now, noting that (Proposition 5.1(b)) the r.v. Z = n(M−θ0 )/σ is standard normal if H0 is true, we have Pθ0 [l(X) ≤ c] = Pθ0 [Z ≥ t(c)] = α and therefore t(c) is the α-upper quantile of the standard normal distribution. This quantity is found on statistical tables and is frequently denoted by the special symbol zα (we ﬁnd, for instance, for α = 0.05; 0.025; 0.01 – the most commonly adopted values of α – the upper quantiles z0.05 = 1.645, z0.025 = 1.960 and z0.01 = 2.326, respectively). Then, in agreement with Neyman–Pearson’s lemma, it follows that the most powerful test for our hypotheses is deﬁned by the critical region '

σ 1 = x : m ≥ θ0 + zα √ n

( (6.14a)

√ and its power is W(θ1 ) = 1 − β = Pθ1 {M ≥ θ0 + zα σ/ n}. This quantity √ can be obtained by noting that under the alternative hypothesis the r.v. n(M − θ1 )/σ is standard normal. Consequently, W(θ1 ) equals√the r-upper quantile of the standard normal distribution, where r = zα − n(θ1 − θ0 )/σ . Since r < zα the area (under the standard normal pdf) to the right of r is greater than the area to the right of zα – which, by deﬁnition, equals α. This shows that, as expected, the test is unbiased. As a numerical example, suppose that we ﬁx a signiﬁcance level α = 0.025 and we wish to test the simple hypotheses H0 : θ = 15.0; H1 : θ = 17.0 knowing that the standard deviation of the underlying normal population is

232 Mathematical statistics σ = 2.0. Suppose further that we carry out n = 20 measurements leading to m = 16.2. Since the rejection region in this case is '

2.0 1 = m ≥ 15.0 + 1.96 √ 20

( = {m ≥ 15.88}

and m = 16.2 falls in it, we reject the null hypothesis and accept H1 . Moreover, we can calculate the power of the test noting that r = −2.51 so that the corresponding upper quantile is 0.994 = W(θ1 ) and the probability of a type II error is β = 1 − 0.994 = 0.006. Following the same line of reasoning as above, it is easy to determine that the rejection region in the case θ1 < θ0 is ' ( σ 1 = x : m ≤ θ0 − zα √ n

(6.14b)

because we get the condition Pθ0 [l(X) ≤ c] = Pθ0 [Z ≤ t(c)] = α thus implying that now t(c) – where t(c) is as in eq. (6.13b) – is the α-lower quantile of the standard normal distribution. Owing to the symmetry of the distribution, this lower quantile is −zα and therefore eq. (6.14b) follows. As a further development of the exercise, consider the following problem: in the case θ1 > θ0 we have ﬁxed the probability of a type I error to a value α, what (minimum) sample size do we need to obtain a probability of type II error smaller than a given value β? The probability of a type II error is √ √ Pθ1 (M < θ0 + zα σ/ n) = Pθ1 {Z < zα − n(θ1 − θ0 )/σ } √ where Z is the standard normal√r.v. Z = n(M − θ1 )/σ . The desired upper limit β is obtained when zα − n(θ1 − θ0 )/σ equals the β-lower quantile of the standard normal distribution. If we denote this lower quantile by qβ we get n=

σ 2 (zα − qβ )2 (θ1 − θ0 )2

(6.15)

and consequently n˜ = [n] + 1 (the square brackets denote the integer part of the number) is the minimum required sample size. So, for instance, taking the same numerical values as above for α, θ0 , θ1 , σ , suppose we want a maximum probability of type II error β = 0.001. Then, the minimum sample size is n˜ = 26 (because z0.025 = 1.96, q0.001 = −3.09 and eq. (6.15) gives n = 25.5). Example 6.1(b) Consider now a normal population with known mean µ and unknown variance σ 2 = θ 2 . Somehow we know that the variance is

The test of statistical hypotheses θ02

θ12

θ12

either or (with of simple hypotheses

>

θ02 )

233

and the scope of the analysis is to test the pair

H0 : θ 2 = θ02

(6.16)

H1 : θ 2 = θ12

Again, we use Neyman–Pearson’s lemma to obtain the most powerful test for the case at hand. Since

n 1 1 1 2 (xi − µ) exp − − 2 l(x) = 2 θ02 θ1 i=1 n

n θ12 − θ02 θ1 xi − µ 2 = exp − θ0 θ0 2θ12 i=1

θ1 θ0

n

the inequality deﬁning the rejection region (6.9) holds if

n xi − µ 2 θ0

i=1

≥

2θ12 θ12

− θ02

[n log(θ1 /θ0 ) − log c] ≡ t(c)

(6.17)

Under the hypothesis H0 , each one of the n independent r.v.s Yi = [(Xi − µ)/θ0 ]2 is distributed according to the χ 2probability law with one degree of freedom. Consequently, the sum Y = Yi has a χ 2 distribution with n degrees of freedom and the relation Pθ0 [l(X) ≤ c] = Pθ0 [Y ≥ t(c)] = α means that t(c) must be the α-upper quantile of this distribution. Then, denoting 2 , the rejection region for the test is this quantile by the symbol χα;n 1 = x :

n

6 2

(xi − µ) ≥

2 θ02 χα;n

(6.18a)

i=1

As a numerical example, suppose we ﬁx a signiﬁcance level α = 0.05 and we wish to test the hypotheses H0 : θ 2 = 3.0; H1 : θ 2 = 3.7 for a normal population with mean µ = 18. If we carry out an experiment consisting of, say, 15 measurements x = (x1 , . . . , x15 ), our rejection region will be 1 = x :

15 i=1

6 2 (xi − 18)2 ≥ 3.0χα;n

= x:

15 i=1

6 (xi − 18)2 ≥ 74.988

234 Mathematical statistics 2 because from statistical tables we get the upper quantile χ0.05;15 = 24.996. 2 2 It is left to the reader to show that in the case θ1 < θ0 the rejection region is

1 = x :

n

6 2

(xi − µ) ≤

2 θ02 χ1−α;n

(6.18b)

i=1 2 where χ1−α;n is the (1 − α)-upper quantile (or, equivalently, the α-lower quantile) of the χ 2 distribution with n degrees of freedom. So, the basic idea of Proposition 6.1 is rather intuitive: since the likelihoodratio statistics (6.10) can be considered as a relative measure of the ‘weight’ of the two hypotheses , l(x) > 1 suggests that the observed data support the null hypothesis while the relation l(x) < 1 tends to imply the opposite conclusion and the speciﬁc value of l(x) – that is c in eq. (6.9) – below which we reject H0 depends on α, that is, the risk we are willing to take of making a type I error. In this light, therefore, it is evident that nothing would change if, as some authors do, one deﬁned the likelihood-ratio as l(X) = L(X; θ1 )/L(X; θ0 ) and considered the rejection region 1 = {l(x) ≥ c} with Pθ0 [l(X) ≥ c] = α. When the probability distributions are discrete the same line of reasoning leads to the most powerful test for the simple hypotheses (6.8). Discreteness, however, often introduces one minor inconvenience. In fact, since the likelihood-ratio statistic takes on only discrete values, say l1 , l2 , . . . , lk , . . ., it may not be possible to satisfy the condition Pθ0 [l(X) ≤ c] = α exactly. The following example will clarify this situation.

Example 6.2 At the signiﬁcance level α, suppose that we want to test the simple hypotheses (6.8) (with θ1 > θ0 ) on the unknown parameter p = θ of a binomial model. Deﬁning y = ni=1 xi we have l(x) =

θ0 θ1

y

1 − θ0 1 − θ1

n−y

and l(x) ≤ c if

1 − θ0 −1 1 − θ0 θ1 + log − log c ≡ t(c) n log y ≥ log θ0 1 − θ1 1 − θ1 Under the null hypothesis, the r.v. Y = X1 + · · · + Xn – being the sum of n binomial r.v.s – is itself binomially distributed with parameter θ0 , and in order to meet the condition Pθ0 [l(X) ≤ c] = Pθ0 [Y ≥ t(c)] = α exactly there should exist an (integer) index k = k(α) such that n n m θ (1 − θ0 )n−m = α m 0

m=k(α)

(6.19)

The test of statistical hypotheses

235

If such an index does exist – a rather rare occurrence indeed – the test attains the desired signiﬁcance level and the rejection region is 1 = x : y =

n

6 xi ≥ k(α)

(6.20)

i=1

However, the most common situation by far is the case in which eq. (6.19) is not satisﬁed exactly but we can ﬁnd an index r = r(α) such that n n m θ (1 − θ0 )n−m < α < m 0

α ≡

m=r(α)

n m=r(α)−1

n m θ (1 − θ0 )n−m ≡ α m 0 (6.21)

At this point we can deﬁne the rejection region as (a) 1 = {x : y ≥ r(α)} or as (b) 1 = {x : y ≥ r(α) − 1}, knowing that in both cases we do not attain the desired level α but we are reasonably close to it. In case (a), in fact, the actual signiﬁcance level α is slightly lower than α (r(α) is the minimum index satisfying the left-hand side inequality of (6.21)) while in case (b) the actual signiﬁcance level α is slightly greater than α (r(α)−1 is the maximum index satisfying the right-hand side inequality of (6.21)). Also, in terms of power we have n n m 1−β = θ (1 − θ1 )n−m < m 1

m=r(α)

n

m=r(α)−1

n m θ (1 − θ1 )n−m = 1 − β m 1

and, as expected, β > β . In the two cases, respectively, Proposition 6.1 guarantees that these are the most powerful tests at levels α and α . Besides the cases (a) and (b) – which in most applications will do – a third possibility called ‘randomization’ allows the experimenter to attain the desired level α exactly. Suppose that we choose the rejection region (b) associated to a level α > α and deﬁned in terms of the index s(α) ≡ r(α)−1. Under the null hypothesis, let us call P0 the probability of the event Y = s(α), that is, P0 ≡ Pθ0 {Y = s(α)} =

n θ s(α) (1 − θ0 )n−s(α) s(α) 0

(which, on the graph of the PDF F0 (x), is the jump F0 (r) − F0 (s)) and let us introduce the ‘critical (or rejection) function’ g(x) deﬁned as g(x) =

⎧ ⎨

1, y > s(α) (P0 + α − α )/P0 , y = s(α) ⎩ 0, y < s(α)

(6.22)

236 Mathematical statistics Then we reject H0 if y > s(α), we accept it if y < s(α) and, if y = s(α), we reject it with a probability (P0 + α − α )/P0 – or, equivalently, accept it with the complementary probability (α − α)/P0 . This means that if y = s(α) we have to set up another experiment with two possible outcomes, one with probability (P0 + α − α )/P0 and the other with probability (α − α)/P0 ; we reject H0 if the ﬁrst outcome turns out, otherwise we accept it. The probability of type I error of this randomized test (i.e. its signiﬁcance level) is obtained by taking the expectation of the critical function and we get, as expected P(H1 |H0 ) = Eθ0 [g(x)] = Pθ0 {y > s(α)} +

P0 + α − α Pθ0 {y = s(α)} P0

= α − P0 +

P0 + α − α P0 = α P0

The reader is invited to: (a) show that the case θ0 > θ1 leads to the rejection region 1 = {x : y ≤ r(α)} where, taking r(α) as the maximum index satisfying the inequality α ≡

r(α) n m=0

m

θ0m (1 − θ0 )n−m ≤ α

(6.23)

the attained signiﬁcance level α is slightly lower than α (unless we are so lucky to have the equal sign in (6.23)); (b) work out the details of randomization for this case. So, in the light of Example 6.2 we can make the following general considerations on discrete cases: (i) Carrying out a single experiment, discreteness generally precludes the possibility of attaining the speciﬁed signiﬁcance level α exactly. Nonetheless we can ﬁnd a most powerful test at a level α < α or at a level α > α. (ii) At this point, we can either be content of α (or α , whichever is our choice) or – if the experiment leads to a likelihood-ratio value on the border between the acceptance and rejection regions – we can ‘randomize’ the test in order to attain α. In the ﬁrst case we lack the probability α − α while we have a probability α − α in excess in the second case. Broadly speaking, randomization compensates for this part by adding a second experimental stage.

The test of statistical hypotheses

237

(iii) This second stage can generally be carried out by looking up a table of random numbers. So, referring to Example 6.2, suppose that we get (P0 + α − α )/P0 = 0.65 and before the experiment we have arbitrarily selected a certain position (say, 12th from the top) of a certain column at a certain page of a two-digit random numbers table. If that number lies between 00 and 64 we reject H0 and accept it otherwise. By so doing, we have performed the most powerful test of size α on the simple hypotheses (6.8). 6.3.2

A few notes on sequential analysis

So far we have considered the sample size n as a number ﬁxed in advance. Even at the end of Example 6.1(a), once we have chosen the desired values of α and β, eq. (6.15) shows that n can be determined before the experiment is carried out. A different approach due to Abraham Wald and called ‘sequential analysis’ leads to a decision on the null hypothesis without ﬁxing the sample size in advance but by considering it as a random variable which depends on the experiment’s outcomes. It should be pointed out that sequential analysis is a rather broad subject worthy of study in its own right (see, for instance, Wald’s book [22]) but here we limit ourselves to some general comments relevant to our present discussion. As in the preceding section, suppose that we wish to test the two simple hypotheses (6.8). For k = 0, 1 let Lkm ≡ L(x1 , . . . , xm ; θk ) =

m

fk (xi )

(6.24)

i=1

be the two likelihood functions L0m , L1m under the hypothesis H0 , H1 , respectively, after m observations (i.e. the realization x1 , . . . , xm ). Then, the general idea of Wald’s sequential test is as follows : (i) we appropriately ﬁx two positive numbers r, R (r < 1 < R), (ii) we continue testing as long as the likelihood ratio lm ≡ L0m /L1m lies between the two limits r, R and (iii) terminate the process for the ﬁrst index which violates one of the inequalities r

θ0 , (b) H1 : θ < θ0 or (c) H1 : θ = θ0 and one speaks of one-sided alternative in cases (a) and (b) – right- and left-sided, respectively – and of two-sided alternative in case (c). With composite hypotheses , a uniformly most powerful (ump) test exists only for some special classes of problems but many of these, fortunately, occur quite often in practice. So, for instance, many statistical models for which there is a sufﬁcient statistic T (for the parameter θ under test so that eq. (5.52) holds) have a monotone (in T) likelihood ratio; for these models it can be shown (see Ref. [10] or [15]) that a ump test to verify H0 : θ = θ0 against a one-sided alternative does exist. This ‘optimal’ test, moreover, coincides with the Neyman–Pearson’s test for H0 : θ = θ0 against an arbitrarily ﬁxed alternative H1 , where H1 is in the form (a) or (b). Even more, the ﬁrst test is also the ump test for the doubly composite case H0 : θ ≤ θ0 ; H1 : θ > θ0 while the second is the ump test for H0 : θ ≥ θ0 ; H1 : θ < θ0 . In spite of all these interesting and important results , it is not our intention to enter into such details and we refer the interested reader to more specialized literature. Here, after some examples of composite hypotheses cases , we will limit ourselves to the description of the general method called ‘likelihood ratio test’ which – although not leading to ump tests in most cases – has a number of other desirable properties.

240 Mathematical statistics Example 6.3(a) If we wish to test the hypotheses H0 : θ = θ0 ; H1 : θ > θ0 for the mean of a normal model with known variance σ 2 , we can follow the same line of reasoning of Example 6.1(a) and obtain the rejection region (6.14a). Since this rejection region does not depend on the speciﬁc value θ1 against which we compare H0 (provided that θ1 > θ0 ), it turns out that this is the uniformly most powerful test for the case under investigation and, as noted above, for the pair of hypotheses H0 : θ ≤ θ0 ; H1 : θ > θ0 as well. Similar considerations apply to the problem H0 : θ = θ0 ; H1 : θ < θ0 and we can conclude that the rejection region (6.14b) provides the ump test for this case and for H0 : θ ≥ θ0 ; H1 : θ < θ0 . Example 6.3(b) Considering the normal model of Example 6.1(b) – that is, known mean and unknown variance σ 2 = θ 2 – it is now evident that the rejection region (6.18a) provides the ump test for the problem H0 : θ 2 ≤ θ02 ; H1 : θ 2 > θ02 while (6.18b) applies to the case H0 : θ 2 ≥ θ02 ; H1 : θ 2 < θ02 . In all the cases above, the probability of a type II error β (and therefore the power) will depend on the speciﬁc value of the alternative. Often, in fact, one can ﬁnd graphs of β plotted against an appropriate variable with the sample size n as a parameter. These graphs are called operating characteristic curves (OC curves ) and the variable on the abscissa axis depends on the type of test. So, for instance, the OC curve for the ﬁrst test of Example 6.3(a) plots β versus (θ1 − θ0 )/σ for some values of n. Fig. 6.1 is one such graph for α = 0.05 and the three values of sample size n = 5, n = 10 and n = 15. As it should be expected, β decreases as the difference θ1 − θ0 increases and, for a ﬁxed value of this quantity, β is lower for larger sample sizes. Similar curves can generally be drawn with little effort for the desired sample size by using widely available software packages such as, for instance,

Beta (prob. of type II error)

1.2 n=5 n =10 n =15

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.40

0.80

1.20 (1 – 0)/

1.60

Figure 6.1 One-sided test (size = 0.05 − H1 : θ > θ0 ).

2.00

2.40

The test of statistical hypotheses Excel® ,

Matlab®

241

etc. The reader is invited to do so for the cases of

Example 6.3(b). Example 6.3(c) Referring back to Example 6.3(a) – normal model with known variance – let us make some considerations on the two-sided case H0 : θ = θ0 ; H1 : θ = θ0 . At the signiﬁcance level α, the rejection regions for the one-sided alternatives H1 : θ > θ0 and H1 : θ < θ0 are given by eqs (6.14a) and (6.14b), respectively. If we conveniently rewrite these equations as √ n(m − θ0 )/σ ≥ zα } √ − 1 = {x : n(m − θ0 )/σ ≤ −zα } + 1 = {x :

(6.29)

we may think of specifying the rejection region 1 of the two-sided test (at + the level α) as 1,α = − 1,a ∪ 1,b , where a, b are two numbers such that a + b = α. Moreover, intuition suggests to take a ‘symmetric’ region by choosing a = b = α/2 thus obtaining √ ' ( n ˜ 1 = x : |m − θ0 | ≥ zα/2 σ

(6.30)

which, in other words , means that we reject the null hypothesis when θ0 is sufﬁciently far – on one side or the other – from the sample mean m. As before, the term ‘sufﬁciently far’ depends on the risk involved in rejecting a true null hypothesis (or, equivalently, accepting a false alternative). We do not do it here but these heuristic considerations leading to (6.30) can be justiﬁed on a more rigorous basis showing that, for the case at hand, eq. (6.30) is a good choice because it deﬁnes the ump test among the class of unbiased tests. In fact, it turns out that a ump test does not exist for this case because (at the level α) the two tests leading to (6.29) can be considered in their own right as tests for the alternative H1 : θ = θ0 . In this light, we already know that (i) the + 1 -test is the most powerful in the region θ > θ0 (ii) the − -test is the most powerful for θ < θ0 and (iii) both their powers take on 1 the value α at θ = θ0 . However, as tests against H1 : θ = θ0 , they are biased. In fact, the power W + (θ ) of the ﬁrst test is rather poor (i.e. low and such that W + (θ) < α) for θ < θ0 and the same holds true for W − (θ ) when θ > θ0 ˜ ) the power of the test (6.30), we have the inequalities so that, calling W(θ ˜ ) < W − (θ ), W + (θ) < α < W(θ

θ < θ0

˜ ) < W + (θ ), W − (θ) < α < W(θ

θ > θ0

˜ 0 ) = α. The fact that for two-sided while, clearly, W − (θ0 ) = W + (θ0 ) = W(θ alternative hypothesis there is no ump test but there sometimes exists a ump

242 Mathematical statistics unbiased test is more general than the special case considered here and the interested reader can refer, for instance, to [15] for more details. As pointed out above, the likelihood ratio method is a rather general procedure used to test composite hypotheses of the type (6.2). In general, it does not lead to ump tests but gives satisfactory results in many practical problems. As before, let X = (X1 , . . . , Xn ) be a random sample and let the (absolutely continuous) model be expressed in terms of the pdf f (x). The likelihood ratio statistic is deﬁned as λ(X) =

supθ∈0 L(X; θ ) supθ∈ L(X; θ )

=

L(θˆ0 ) L(θˆ )

(6.31)

where θˆ0 is the maximum likelihood (ML) estimate of the unknown parameter θ when θ ∈ 0 and θˆ is the ML estimate of θ over the entire parameter space . Deﬁnition 6.31 shows that 0 ≤ λ ≤ 1 because we expect λ to be close to zero when the null hypothesis is false and close to unity when H0 is true. In this light, the rejection region is deﬁned by 1 = {x : λ(x) ≤ c}

(6.32)

where the number c is determined by the signiﬁcance level α and it is such that sup Pθ {λ(X) ≤ c} = α

θ∈0

(6.33)

which amounts to the condition P(H1 |H0 ) = Pθ {λ(X) ≤ c} ≤ α for all θ ∈ 0 (recall the deﬁnition of power (6.4) and eq. (6.5)). It is clear at this point that the likelihood ratio test generalizes Neyman–Pearson’s procedure to the case of composite hypotheses and reduces to it when both hypotheses are simple. If the null hypothesis is simple – that is, of the form H0 : θ = θ0 – the numerator of the likelihood ratio is simply L(θ0 ) and, since 0 contains only the single element θ0 , no ‘sup’ appears both at the numerator of (6.31) and in eq. (6.33). Example 6.4(a) In Example 6.3(a) we have already discussed the normal model with known variance when the test on the mean µ = θ is of the form H0 : θ ≤ θ0 ; H1 : θ > θ0 . Let us now examine it in the light of the likelihood ratio method. For this case, clearly, = R, 0 = (−∞, θ0 ] and 1 = (θ0 , ∞). The denominator of (6.31) is L(M) because the sample mean M is the ML estimator of the mean (recall Section 5.5) over the entire parameter space. On the other hand, it is not difﬁcult to see that the numerator of (6.31) is L(θ0 ) so that n λ(X) = exp − 2 (M − θ0 )2 2σ

(6.34)

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243

and the inequality in (6.32) is satisﬁed if : |m − θ0 | ≥

2σ 2 log(1/c) = t(c) n

where, as usual, m is the realization of M. Since m > θ0 in the rejection region, the condition (6.33) reads √ '√ ( n(M − θ ) n(θ0 − θ + t(c)) ≥ σ σ θ≤θ0 √

nt(c) =P Z≥ =α σ

sup P{M ≥ θ0 + t(c)} = sup P

θ≤θ0

√ because in 1 the r.v. Z = n(M − θ )/σ is standard normal and the ‘sup’ of the probability above is attained when θ = θ0 . The conclusion is that, as expected, the rejection region is given by eq. (6.14), that is, the same as for the simple case H0 : θ = θ0 ; H1 : θ = θ1 with θ1 > θ0 . Also, from the considerations above we know that eq. (6.14a) deﬁnes the ump for the problem at hand. Example 6.4(b) The reader is invited to work out the details of the likelihood ratio method for the normal case in which the hypotheses on the mean are H0 : θ = θ0 ; H1 : θ = θ0 and the variance is known. We have in this case = R, 0 = {θ0 } and 1 = (−∞, θ0 ) ∪ (θ0 , ∞) and, as above, λ(X) = L(θ0 )/L(M) so that eq. (6.34) still applies. Now, however, no √‘sup’ needs to be taken in eq. (6.33) which, in turn, becomes P{|Z| ≥ n t(c)/σ } = α thus leading to the rejection region (6.30) where, we recall, zα/2 denotes the α/2-upper quantile of the standard normal distribution. So, for instance, if α = 0.01 we have zα/2 = z0.005 = 2.576 while for a 10 times smaller probability of type I error – that is, α = 0.001 – we ﬁnd in tables the value zα/2 = z0.0005 = 3.291. The fact that (a) testing hypotheses on the mean of a normal model with known variance leads to rejection regions where the quantiles of the standard normal distribution appear; (b) testing hypotheses on the variance (Examples 6.1(b) and 6.3(b)) of normal model with known mean brings into play the quantiles of the χ 2 distribution with n degrees of freedom may suggest a connection between parametric hypothesis testing and interval estimation problems (Section 5.6). The connection, in fact, does exist and is

244 Mathematical statistics rather strong. Before considering the situation from a general point of view, we give two more examples that conﬁrm this state of affairs. Example 6.5(a) Consider a normal model with the same hypotheses on the mean as in Example 6.4(b) but with the important difference that the variance σ 2 is now unknown (note that in this situation the null hypothesis H0 : θ = θ0 is not simple because it does not uniquely determine the probability distribution). In this case, the parameter space = R × R+ is divided into the two sets 0 = {θ0 } × R+ and 1 = [(−∞, θ0 ) ∪ (θ0 , ∞)] × R+ . Under the null hypothesis θ = θ0 the maximum of the likelihood function is attained when σ 2 = σ02 = n−1 i (xi − θ0 )2 and the numerator of (6.31) is L(θ0 , σ02 ) = (2π σ02 e)−n/2 2 2 −n/2 Similarly, the denominator of (6.31) 2is given by L(M, S ) = (2π S e) 2 −1 because M and S = n i (Xi − M) , respectively, are the ML estimators of the mean and the variance. Consequently, if s2 is the realization of the sample variance S2 , we get

λ=

σ02 s2

−n/2

t2 = 1+ n−1

−n/2 (6.35)

√ where we set t = t(x) = n − 1(m − θ0 )/s and the second relation is easily obtained by noting that σ02 = s2 + (m − θ0 )2 . Equation (6.35) shows that there is a one-to-one correspondence between λ and t 2 and therefore the inequality in (6.32) is equivalent to |t| ≥ c (where c is as appropriate). Since the r.v. t(X) is distributed according to a Student distribution with n−1 degrees of freedom, the boundary c of the rejection region (at the signiﬁcance level α) must be the α/2-upper quantile of this distribution. Denoting this upper quantile by tα/2;n−1 we have 6 √ n−1 |m − θ0 | ≥ tα/2;n−1 1 = x : s

(6.36)

Alternatively, if one prefers to do so, 1 can be speciﬁed in terms of the ¯2 (1 − α/2)-lower quantile √ and/or using √ the unbiased estimator S instead of 2 S (and recalling that n − 1/s = n/¯s). As a numerical example, suppose we carried out n = 10 trials to test the hypotheses H0 : θ = 35; H1 : θ = 35 at the level α = 0.01. The rejection region is then 1 = {x : (3/s)|m − 35| ≥ 3.25} because the 0.005-upper quantile of the Student distribution with 9 degrees of freedom is t0.005; 9 = 3.250. So, if the experiment gives, for instance, m = 32.6 and s = 3.3 we fall outside 1 and we accept the null hypothesis at the speciﬁed signiﬁcance level.

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245

In the lights of the comments above, the conclusion is as expected: testing the mean of a normal model with unknown variance leads to the appearance of the Student quantiles, in agreement with Example 5.9(b) where, for the same model, we determined a conﬁdence interval for the mean (eq. (5.76)). Example 6.5(b) At this point it should not be surprising if we say that testing the variance of a normal model with unknown mean involves the quantiles of the χ 2 distribution with n − 1 degrees of freedom (recall Example 5.10(b)). It is left to the reader to use the likelihood ratio method to determine that the rejection region for the hypotheses H0 : θ 2 = θ02 ; H1 : θ 2 = θ02 on the variance is 6 , , s¯2 s¯2 2 2 1 = x : 0 ≤ n − 1 2 ≤ χ1−α/2;n−1 ∪ n − 1 2 ≥ χα/2;n−1 θ0 θ0 (6.37)

2 2 , respectively, are the (1−α/2)-upper quantile and χα/2;n−1 where χ1−α/2;n−1 and the α/2-upper quantile of the χ 2 distribution with n − 1 degrees of freedom. So, for instance, if n = 10 and we are testing at the level α = 0.05 2 2 2 2 we ﬁnd χ1−α/2;n−1 = χ0.975;9 = 2.70 and χα/2;n−1 = χ0.025;9 = 19.023.

Therefore, we will accept the null hypothesis H0 if 2.70 < 3¯s2 /θ02 < 19.023. 6.3.4

Complements on parametric hypothesis testing

In addition to the main ideas of parametric hypotheses testing discussed in the preceding sections, some further developments deserve consideration. Without claim of completeness, we do this here by starting from where we left off in Section 6.3.3: the relationship with conﬁdence intervals estimation. 6.3.4.1

Parametric tests and conﬁdence intervals

Let us ﬁx a signiﬁcance level α. If we are testing H0 : θ = θ0 against the composite alternative H1 : θ = θ0 , the resulting rejection region 1 will depend, for obvious reasons , on the value θ0 and so will the acceptance region 0 = C 1 . As θ0 varies in the parameter space we can deﬁne the family of sets 0 (θ ) ⊂ , where each 0 (θ ) corresponds to a value of θ. On the other hand, a given realization of the sample x will fall – or will not fall – in the acceptance region depending on the value of θ under test and those values of θ such that x does fall in the acceptance region deﬁne a subset G ⊂ of the parameter space. By letting x free to vary in , therefore, we can deﬁne the family of subsets G(x) = {θ : x ∈ 0 (θ )} ⊂ . The consequence of this (rather intricate at ﬁrst sight) construction of subsets in the sample and parameter space is the fact that the events {X ∈ H0 (θ )} and

246 Mathematical statistics {θ ∈ G(X)} are equivalent, and since the probability of the ﬁrst event is 1 − α so is the probability of the second. Recalling Section 5.6, however, we know that this second event deﬁnes a conﬁdence interval for the parameter θ. By construction, the conﬁdence level associated to this interval is 1 − α = γ . The conclusion, as anticipated, is that parametric hypothesis testing and conﬁdence interval estimation are two strictly related problems and the solution of one leads immediately to the solution of the other. Moreover, a ump test – if it exists – corresponds to the shortest conﬁdence interval and vice versa. Having established the nature of the connection between the two problems, we can now reconsider some of the preceding results from this point of view. Let us go back to Example 6.3(c) where the rejection region is given by eq. (6.30). This relation implies that the ‘acceptance’ subsets 0 (θ ) have the form ( ' σ σ 0 (θ) = x : m − √ zα/2 < θ < m + √ zα/2 n n

(6.38)

Those x such that x ∈ 0 (θ ) correspond to the values of θ which satisfy the inequality in (6.38) and these θ, in turn, deﬁne the corresponding set √ G(x). From the discussion above it follows that (M ± zα/2 σ/ n) is a 1 − α conﬁdence interval for the mean of a normal model with known variance. This is in agreement with eq. (5.75) of Example 5.9(a) because the α/2-upper quantile zα/2 (of the standard normal distribution) is the (1 − α/2)-lower quantile and, since 1 − α = γ , this is just the (1 + γ )/2-lower quantile (which was denoted by the symbol c(1+γ )/2 in Example 5.9(a)). Similarly, for the same statistical model we have seen that the rejection region to test H0 : θ = θ0 ; H1 : θ > θ0 is given by√(6.14a). This implies that the acceptance sets are 0 (θ ) = {x : m < θ +zα σ/√ n} and the corresponding sets G(x) are given by G(x) = {θ : θ > m − zα σ/ n}. The conclusion is that

σ M − zα √ , +∞ n

(6.39a)

is a lower (1 − α)-conﬁdence (one-sided) interval for the mean. The interval (6.39a) is called ‘lower’ because only the lower limit for θ (recall eq. (5.69a)) is speciﬁed; by the same token, it is evident that we can use the rejection region (6.14b) to construct the upper (1 − α)-conﬁdence interval

σ −∞, M + zα √ n

(6.39b)

The argument, of course, works in both directions and we can, for instance, start from the γ -CI (5.76) for the mean of a normal model with unknown

The test of statistical hypotheses

247

variance to write the acceptance region for the test H0 : θ = θ0 ; H1 : θ = θ0 as √ ' ( n(m − θ0 ) 0 = x : −t(1+γ )/2;n−1 < < t(1+γ )/2;n−1 s¯ from which we get the rejection region 1 = C 0 of eq. (6.36) by noting that √ √ (i) n − 1/s = n/¯s and (ii) (1 + γ )/2 = 1 − α/2 so that the Student lower quantiles of (5.76) are the 1 − (1 − α/2) = α/2 (Student) upper quantiles of eq. (6.36). At this point it is left to the reader to work out the details of the parametric hypothesis counterparts of Examples 5.11 (a–c) by determining the relevant acceptance and rejection regions. 6.3.4.2

Asymptotic behaviour of parametric tests

The second aspect we consider is the asymptotic behaviour of parametric tests. As a ﬁrst observation, we recall from Chapter 5 that a number of important sample characteristics are asymptotically normal with means and variances determined by certain population parameters. Consequently, when the test concerns one of these characteristics the ﬁrst thing that comes to mind is, for large samples, to use the normal approximation by replacing any unknown population parameter by its (known) sample counterpart thus obtaining a rejection region determined by the appropriate quantile of the standard normal distribution. This is a legitimate procedure but it should be kept in mind that it involves two types of approximations (i) the normal approximation for the distribution of the characteristic under test and (ii) the use of sample values for the relevant unknown population parameters. So, in practice, it is often rather difﬁcult to know whether our sample is large enough and our test has given a reliable result. As a rule of thumb, n > 30 is generally good enough when we are dealing with means while n > 100 is advisable for variances, medians , coefﬁcients of skewness and kurtosis. For some other ‘less tractable’ characteristics, however, even samples as large as 300 or more do not always give a satisfactory approximation. Let us now turn our attention to Neyman–Pearson’s lemma on simple hypotheses (Proposition 6.1). The boundary value c in eq. (6.9) can only be calculated when we know the distribution of the statistic l(X) under H0 (and, similarly, the probability β of a type II error can be calculated when we know the distribution of l(X) under H1 ). Since this is not always possible, we can proceed as follows. If we deﬁne the r.v.s Yi = log

f0 (Xi ) f1 (Xi )

(6.40)

for i = 1, 2, . . . , n then Sn = Y1 + Y2 + · · · + Yn is the sum of n iid variables. Depending on which hypothesis is true, its mean and variance are

248 Mathematical statistics Eθ0 (Sn ) = na0 and Varθ0 (Sn ) = nσ02 under H0 or Eθ1 (Sn ) = na1 and Varθ1 (Sn ) = nσ12 under H1 , where we called a0 , σ02 and a1 , σ12 the mean and variance (provided that they exist) of the variables Yi under H0 and H1 , respectively. At this point, the CLT √ (Proposition 4.22) tells us that, under H0 , the r.v. Z = (Sn − na0 )/σ0 n is asymptotically standard normal and, since deﬁnition (6.40) implies that the inequality in eq. (6.9) is equivalent to Sn ≤ log c, for sufﬁciently large values of n we can write ' Pθ0

log c − na0 Z≤ √ σ0 n

( =α

(6.41)

which, on the practical side, implies that we have an approximate rejection region: we reject the null hypothesis if the realization of the sample is such that (y1 + y2 + · · · + yn ) − na0 ≤ cα √ σ0 n

(6.42)

where cα is the α-lower quantile of the standard normal distribution. Clearly, the goodness of the approximation depends on how fast the variable Z converges (in distribution) to the standard normal r.v.; if the rate of convergence is slow, a rather large sample is required to obtain a reliable test. In regard to the more general likelihood ratio method, it is convenient to consider the monotone function (X) = −2 log λ(X) of the likelihood ratio λ(X). Provided that the regularity conditions for the existence, uniqueness and asymptotic normality of the ML estimate θˆ of the parameter θ are met (see Sections 5.5 and 5.5.1), it can be shown that the asymptotic rejection region for testing the null hypothesis H0 : θ = θ0 is given by 2 1 = {x : (x) ≥ χ1−α;1 }

(6.43)

2 where χ1−α;1 is the (1 − α)-lower quantile of the χ 2 distribution with one degree of freedom. We do not prove this assertion here but we note that eq. (6.43) is essentially due to the fact that, under H0 , we have

(i) (X) → χ 2 (1)[D]; 2 (ii) Pθ0 {(X) ≥ χ1−α; 1} → α as n → ∞. The result can also be extended directly to the case of a vector, say k-dimensional, parameter q = (θ1 , . . . , θk ) and it turns out that the rejection region is deﬁned by means of the (1−α)-lower quantile of the χ 2 distribution with k degrees of freedom. In addition, the procedure still applies if the null hypothesis speciﬁes only a certain number, say r, of the k components of q. In this case the numerator of (6.31) is obtained by maximizing L with

The test of statistical hypotheses

249

respect to the remaining k − r components and r is the number of degrees of freedom of the asymptotic distribution. Also, another application of the result stated by eq. (6.43) is the construction of conﬁdence intervals for a general parametric model. In fact, eq. (6.43) implies that the (asymptotic) acceptance region is 0 = {x : (x) < χγ2;1 }, where γ = 1 − α. In the light of the relation between hypothesis testing and CIs we have that G(X) = {θ : (X) < χγ2;1 } is an asymptotic γ -CI for the parameter θ because Pθ {θ ∈ G(X)} → γ as n → ∞. In practice G(X) – called a maximum likelihood conﬁdence interval – can be used as an approximate CI when the sample size is large. All these further developments , however – together with the proof of the theorem above – are beyond our scope. For more details the interested reader may refer, for instance, to [1, 2, 10, 13, 19]. 6.3.4.3

The p-value: signiﬁcance testing

A third aspect worthy of mention concerns a slightly different implementation of the hypothesis testing procedure shown in the preceding sections. This modiﬁed procedure is often called ‘signiﬁcance testing’ and is becoming more and more popular because it somehow overcomes the rigidity of hypothesis testing. The main idea of signiﬁcance testing originates from the fact that the choice of the level α is, to a certain extent, arbitrary and the common values 0.05, 0.025 and 0.01 are often used out of habit rather than through careful analysis of the consequences of a type I error. So, instead of ﬁxing α in advance we perform the experiment, calculate the value of the appropriate test statistic and report the so-called p-value (or ‘observed signiﬁcance level’ and denoted by αobs ), deﬁned as the smallest value of α for which we reject the null hypothesis. Let us consider an example. Example 6.6 Suppose that we are testing H0 : θ = 100 against H1 : θ < 100 where the parameter θ is the mean of a normally distributed r.v. with known variance σ 2 = 25. Suppose further that an experiment on a sample of n = 16 products gives the sample mean m = 97.5. Since the rejection region for this case is given by eq. (6.14b), a test at level α = 0.05 (z0.05 = 1.645) leads to reject H0 in favour of H1 and the same happens at the level α = 0.025 (z0.025 = 1.960). If, however, we choose α = 0.01 (z0.01 = 2.326) the conclusion is that we must accept H0 . This kind of situation is illustrative of the rigidity of the method; we are saying in practice that we tolerate 1 chance in 100 of making a type I error but at the same time we state that 2.5 chances in 100 is too risky. One way around this problem is, as noted above, to calculate the p-value and √ move on from there. For the case at hand the relevant test statistic n(M − θ0 )/σ is standard normal and attains the value 4(−2.5)/5 = −2.0 which, we ﬁnd on statistical tables, corresponds to the level αobs = 0.0228.

250 Mathematical statistics This is the p-value for our case, that is, the value of α that will just barely cause H0 to be rejected. In other words, on the basis of the observed data we would reject the null hypothesis for any level α ≥ αobs = 0.0228 and accept it otherwise. Reporting the p-value, therefore, provides the necessary information to the reader to decide whether to accept or reject H0 by comparing this value with his/her own choice of α: if one is satisﬁed with a level α = 0.05 then he/she will not accept H0 but if he/she thinks that α = 0.01 is more appropriate for the case at hand, the conclusion is that H0 cannot be rejected. (As an incidental remark, it should be noted that the calculation of the p-value for this example is rather easy but it may not be so if the test statistic is not standard normal and we must rely only on statistical tables. However, the use widely available software packages has made things much easier because p-values are generally given by the software together with all the other relevant results of the test.) The example clariﬁes the general idea. Basically, this approach provides the desired ﬂexibility; if an experiment results in a low p-value, say αobs < 0.01, then we can be rather conﬁdent in our decision of rejecting the null hypothesis because – had we tested it at the ‘usual’ levels 0.05, 0.025 or 0.01 – we would have rejected it anyway. Similarly, if αobs > 0.1 any one of the usual testing levels would have led to the acceptance of H0 and we can feel quite comfortable with the decision of accepting H0 . A kind of ‘grey area’, so to speak, is when 0.01 < αobs < 0.1 and, as a rule of thumb, we may reject H0 for 0.01 < αobs < 0.05 and accept it for 0.05 < αobs < 0.1. It goes without saying, however, that exceptions to this rule are not rare and the speciﬁc case under study may suggest a different choice. In addition, we must not forget to always keep an eye on the probability β of a type II error. 6.3.4.4

Closing remarks

To close this section, a comment of general nature is not out of place: in some cases , taking a too large sample size may be as bad an error as taking a too small sample size. The reason lies in the fact as n increases we are able to detect smaller and smaller differences from the null hypothesis (this, in other words, means that that the ‘discriminating power’ of the test increases as n increases) and consequently we will almost always reject H0 if n is large enough. In performing a test, therefore, we must keep in mind the difference between statistical signiﬁcance and practical signiﬁcance, where this latter term refers to both reasonableness and to the nominal speciﬁcations , if any, for the case under study. So, without loss of generality, consider the test of Example 6.6 and suppose that for all practical purposes it would not matter much if the mean of the population were within ± 0.5 units from the value θ0 = 100 under test. If we decided to take a sample of n = 1600 products obtaining, say, the sample mean m = 99.7, we would reject

The test of statistical hypotheses

251

the null hypothesis at α = 0.05, α = 0.025 and α = 0.01 (incidentally, the p-value in this case is αobs = 0.0082). This is clearly unreasonable because the statistically signiﬁcant difference detected by the test (which leads to the rejection of H0 ) does not correspond to a practical difference and we put ourselves in the same ironical situation of somebody who uses a microscope when a simple magnifying glass would do. As stated at the beginning of this section, these complementary notes do not exhaust the broad subject of parametric hypothesis testing. In particular, the various methods classiﬁed under the name ‘analysis of variance’ (generally denoted by the acronym ANOVA) have not been considered. We just mention here that the simplest case of ANOVA consists in comparing the unknown means µ1 , µ2 , . . . , µk of a number k > 2 of normal populations by testing the null hypothesis H0 : µ1 = µ2 = · · · = µk against the alternative that at least one of the equalities does not hold. As we can immediately see, this is an important case of parametric test where – it can easily be shown – pair wise comparison is not appropriate. For a detailed discussion of this interesting topic and of its ramiﬁcations the reader may refer to [6, 7, 9, 12, 18].

6.4

Testing the type of distribution (goodness-of-ﬁt tests)

In the preceding sections we concerned ourselves with tests pertaining to one (or more) unknown parameter(s) of a known distribution law, thus tacitly implying that we already have enough evidence on the underlying probability distribution – that is, normal, Poisson, binomial or other – with which we are dealing. Often, however, the uncertainty is on the type of distribution itself and we would like to give more support to our belief that the sample X = (X1 , . . . , Xn ) is, in fact, a sample from a certain distribution law. In other words, this means that on the basis of n independent observations of a r.v. X with unknown distribution FX (x) we would like to test the null hypothesis H0 : FX (x) = F(x) against H1 : FX (x) = F(x), where F(x) is a speciﬁed probability distribution. Two of the most popular tests for this purpose are Pearson’s χ 2 -test and Kolmogorov–Smirnov test. 6.4.1

Pearson’s χ 2 -test and the modiﬁed χ 2 -test

Let us start with Pearson’s test by assuming at ﬁrst that F(x) is completely speciﬁed – that is, it is not of the form F(x; θ ), where θ is some unknown parameter of the distribution. Now, let D1 , D2 , . . . , Dr be a ﬁnite partition of the space D of possible values of X (i.e. D = ∪rj=1 Dj and Di ∩ Di = ∅ for i = j) and let pj = P(X ∈ Dj |H0 ) be the probabilities of the event {X ∈ Dj } under H0 . These probabilities – which can be arranged in a r-dimensional vector p = (p1 , . . . , pr ) – are known because they depend on F(x) which,

252 Mathematical statistics in turn, is assumed to be completely speciﬁed. In fact, for j = 1, . . . , r we have P(X = xk ) (6.44a) pj = k : xk ∈Dj

if X is discrete and pj = f (x) dx

(6.44b)

Dj

if X is absolutely continuous with pdf f (x) = F (x). Turning now our attention to the sample X = (X1 , . . . , Xn ), let Nj be the r.v. representing the number of its elements falling in Dj so that νj = Nj /n is a r.v. representing the relative frequency of occurrence pertaining to Dj . Clearly, N1 + · · · + Nr = n or, equivalently r

νj = 1

(6.45)

j=1

With these deﬁnitions the hypotheses under test can be re-expressed as H0 : pj = νj and H1 : pj = νj for j = 1, . . . , r, thus implying that we should accept H0 when the sample frequencies Nj are in reasonable agreement with the ‘theoretical’ (assumed) frequencies npj . It was shown by Pearson that an appropriate test statistic for this purpose is T≡

r (Nj − npj )2 j=1

npj

=

r N2 j j=1

npj

−n

(6.46)

for which, under H0 , we have E(T) = r − 1

⎞ ⎛ r 1 ⎝ 1 − r2 − 2r + 2⎠ Var(T) = 2(r − 1) + n pj

(6.47)

j=1

and, most important, as n → ∞ T → χ 2 (r − 1) [D]

(6.48)

Equation (6.48) leads directly to the formulation of Pearson’s χ 2 goodnessof-ﬁt test: at the signiﬁcance level α, the approximate rejection region to test

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253

the null hypothesis H0 : pj = νj is given by 2 1 = {x : t(x) ≥ χ α;r−1 }

(6.49)

2 is the α-upper where t(x) is the sample realization of the statistic T and χ α;r−1 2 quantile of the χ distribution with r − 1 degrees of freedom. Although we do not prove the results (6.47) and (6.48) – the interested reader can refer to [4] or [10] – a few comments are in order:

(i) Explicitly, t(x) = j (nj − npj )2 /npj where nj (j = 1, . . . , r) is the realization of the r.v. Nj ; in other words, once we have carried out the experiment leading to the realization of the sample x = (x1 , . . . , xn ), n1 is the number of elements of x whose values fall in D1 , n2 is the number of elements of x whose values fall in D2 , and so on. (ii) The rejection region (6.49) is approximate because eq. (6.48) is an asymptotic relation; similarly, the equation P(T ∈ 1 |H0 ) = α deﬁning the probability of a type I error is strictly valid only in the limit of n → ∞. However, the quality of the approximation is generally rather good for n ≥ 50. For better results, many authors recommend to choose the Dj so that npj ≥ 5 for all j, or, at least, for more than 80–85% of all j (since we divide by npj in calculating T, we do not want the terms with the smaller denominators to ‘dominate’ the sum(6.46)). (iii) On the practical side, the choice of the Dj – which, for a onedimensional random variable, are non-overlapping intervals of the real line – plays an important role. Broadly speaking, they should not be too few and they should not be too many, that is, r should be neither too small nor too large. A possible suggestion (although in no way a strict rule) is to use the formula 0.2 2(n − 1)2 ∼ r=2 zα2

(6.50)

where zα is the α-upper quantile of the standard normal distribution. So, for the most common levels of signiﬁcance 0.05, 0.025 and 0.01 we have r ∼ = 1.883(n − 1)0.4 , r ∼ = 1.755(n − 1)0.4 and r ∼ = 1.639(n − 1)0.4 , respectively. Moreover, in order to comply with the indicative rule of point (ii) – that is, npj ≥ 5 for almost all j – adjacent end intervals (which cover the tails of the assumed distribution) are sometimes regrouped to ensure that the minimum absolute frequency is 5 or, at least, not much smaller than 5. Regarding the width of the intervals, it is common practice to choose equal-width intervals, although this requirement is not necessary. Some

254 Mathematical statistics authors, in fact, prefer to select the intervals so that the expected frequencies will be the same in each interval; excluding a uniform distribution hypothesis, it is clear that this choice will result in different interval widths. Example 6.7(a) Suppose that in n = 4000 independent trials the events A1 , A2 , A3 have been obtained n1 = 1905, n2 = 1015 and n3 = 1080 times, respectively. At the level α = 0.05 we want to test the null hypothesis H0 : p1 = 0.5; p2 = p3 = 0.25, where pj = P(Aj ). In this intentionally simple example the assumed distribution is discrete and np1 = 2000, np2 = np3 = 1000. Since the sample realization of the statistic 2 T is t = 11.14 and it is higher than the 0.05-upper quantile χ 0.05;2 = 5.99 we fall in the rejection region (6.49) and therefore, at the level α = 0.05, we reject the null hypothesis. Example 6.7(b) In n = 12000 tosses of a coin Pearson obtained n1 = 6019 heads and n2 = 5981 tails. Let us check at the levels 0.05 and 0.01 if this result is consistent with the hypothesis that he was using a fair coin (i.e. H0 : p1 = p2 = 0.5). We get now t = 0.120 and this value is lower than 2 2 both upper quantiles χ 0.05;1 = 3.841 and χ 0.01;1 = 6.635. Since we fall in the acceptance region in both cases, we accept H0 and infer that the data agree with the hypothesis of unbiased coin. The study of the power W of Pearson’s χ 2 test when H0 is not true is rather involved, also in the light of the fact that W cannot be computed unless a speciﬁed alternative H1 is considered. Nonetheless, an important result is worthy of mention: for every ﬁxed set of probabilities p¯ = p the power ¯ tends to unity as n → ∞ [10]. This, in words, is expressed function W(p) by saying that the test is ‘consistent’ (not to be confused with consistency for an estimator introduced in Section 5.4) and means that, under any ﬁxed alternative H1 , the probability 1 − β of rejecting the null hypothesis when it is false tends to 1 as n increases. It is evident that consistency, for any statistical hypothesis test in general, is a highly desirable property. A modiﬁed version of the χ 2 goodness-of-ﬁt test can be used even when the assumed probability distribution F(x) contains some unknown parameters, that is, it is of the form F(x; q) where q = (θ1 , . . . , θk ) is a set of k unknown parameters. In this case the null hypothesis is clearly composite – in fact, it identiﬁes a class of distributions and not one speciﬁc distribution in particular – and the statistic T itself will depend on q through the probabilities pj , that is, T = T(q) where T(q) =

r [Nj − npj (q)]2 j=1

npj (q)

=

r Nj2 j=1

npj (q)

−n

(6.51)

The test of statistical hypotheses

255

So, if T is the appropriate test statistic for the case at hand – and it turns out that, in general, it is – we must ﬁrst eliminate the indeterminacy brought about by q. One possible solution is to estimate the parameter(s) q by some estimating method and use the estimate q˜ in eq. (6.51). At this point, however, two objections come to mind. First, by so doing the probabilities pj are no longer constants but depend on the sample (in fact, no matter which ˜ thus estimation method we choose, the sample must be used to calculate q), implying that eq. (6.48) will probably no longer hold. Second, if eq. (6.48) does not hold but there exists nonetheless a limiting distribution for T, is this limiting distribution independent on the estimation method used to ˜ obtain q? The way out of this rather intricate situation was found in the 1920s by Fisher who showed that for an important class of estimation methods the χ 2 distribution is still the asymptotic distribution of T but eq. (6.48) must be modiﬁed to ˜ → χ 2 (r − k − 1) [D] T(q)

(6.52)

thus determining that the effect of the k unknown parameters is just a decrease – of precisely k units, one unit for each estimated parameter – of the number of degrees of freedom. Such a simple result does not correspond to a simple proof and the interested reader is referred to Chapter 30 of [4] for the details. There, the reader will also ﬁnd the (rather mild) conditions on the continuity and differentiability of the functions pj (q) for eq. (6.52) to hold. On the practical side, once we have obtained the estimate q˜ by means of an appropriate method – we will come to this point shortly – eq. (6.52) implies that the ‘large-sample’ rejection region for the test is 7 8 2 ˜ ≥ χ α;r−k−1 1 = x : t(x; q)

(6.53)

and, as above, the probability of rejecting a true null hypothesis (type I error) is approximately equal to α. In regard to the estimation method, we can argue that a ‘good’ estimate of q can be obtained by making T(q) as small as possible (note that the smaller is T, the better it agrees with the null hypothesis) so that our estimate q˜ can be obtained by solving for the unknowns θ1 , . . . , θk the system of k equations r nj − npj (nj − npj )2 ∂pj − =0 pj ∂θi 2np2j j=1

(6.54a)

where it is understood that pj = pj (q) and i = 1, 2, . . . , k. This is called the χ 2 minimum method of estimation and its advantage is that, under sufﬁciently general conditions, it leads to estimates that are consistent, asymptotically

256 Mathematical statistics normal and asymptotically efﬁcient. Its main drawback, however, is that ﬁnding the solution of (6.54a) is generally a difﬁcult task. For large values of n, fortunately, it can be shown that the second term within parenthesis becomes negligible and therefore the estimate q˜ can be obtained by solving the modiﬁed, and simpler, equations

r nj − npj ∂pj j=1

pj

∂θi

=

r nj ∂pj j=1

pj

∂θi

=0

(6.54b)

where the ﬁrst equality is due to the condition j pj (q) = 1 for all q ∈ . Equation (6.54b) express the so-called ‘modiﬁed χ 2 minimum method’ which, for large samples, gives estimates with the same asymptotic properties as the χ 2 method of eqs (6.54a) and (6.54b). This asymptotic behaviour of the ‘modiﬁed χ 2 estimates’ is not surprising if one notes that, for the observations grouped by means of the partition D1 , D2 , . . . , Dr , q˜ coincides with the ML estimate qˆ g (the subscript g is for ‘grouped’). In fact, once the partition {Dj }rj=1 has been chosen, the r.v.s Nj are distributed according to the multinomial probability law (eq. (3.46a)) because each observation xi can fall in the interval Dj with probability pj . It follows that the likelihood function of the grouped observations Lg (n1 , . . . , nr ; q) is given by

nj n pj (q) n1 ! · · · nr ! r

Lg (N1 = n1 , . . . , Nr = nr ; q) =

(6.55)

j=1

and the ML estimate qˆ g of q is obtained by solving the system of equations ∂ log Lg /∂θi = 0 (i = 1, 2, . . . , k) which, when written explicitly, coincides with (6.54b). At this point an interesting remark can be made. If we calculate the ML estimate before grouping by maximizing the ‘ungrouped’ likelihood function L(x; q) = f (x1 ; q) · · · f (xn ; q) – which, we note, is different from (6.55) and therefore leads to an estimate qˆ = qˆ g – then qˆ is probably a better estimate than qˆ g because it uses the entire information from the sample (grouping, in fact, leads to a partial loss of information). Furthermore, maximizing L(x; q) is often computationally easier than ﬁnding the solution of eq. (6.54b). Unfortunately, while it is true that (eq. (6.52)) T(qˆ g ) → χ 2 (r − k − 1) [D], it ˆ → χ 2 (r − k − 1) [D] does not hold in general and has been shown that T(q) ˆ is more complicated than the χ 2 distrithe asymptotic distribution of T(q) bution with r−k−1 degrees of freedom. In conclusion, we need to group the data ﬁrst and then estimate the unknown parameter(s); by so doing, Cramer [4] shows that the above results are valid for any set of asymptotically normal and asymptotically efﬁcient estimates of the parameters (however obtained, that is, not necessarily by means of the modiﬁed χ 2 method).

The test of statistical hypotheses

257

Example 6.8 Suppose we want to test the null hypothesis that some observed data come from a normal distribution with unknown mean µ = θ1 and variance σ 2 = θ22 so that q = (θ1 , θ2 ) is the set of k = 2 unknown parameters. For j = 1, . . . , r let the grouping intervals be deﬁned as Dj = (xj−1 , xj ] where, in particular, x0 = −∞, xr = ∞ and xj = x1 + (j − 1) d for j = 1, . . . , r−1, x1 and d being appropriate constants (in other words, the Dj s are a partition of the real line in non-overlapping intervals which, except for the ﬁrst and the last, all have a constant width d). Also, in order to simplify the notation, let 1 g(x; q) = √ exp 2π

−

(x − θ1 )2

2θ22

With these deﬁnitions we have the ‘theoretical’ probabilities pj (q) =

1 θ2

g(x; q) dx

(6.56)

Dj

so that, calculating the appropriate derivatives, the modiﬁed minimal χ 2 estimate’ of q is obtained by solving the system of two equations r nj (x − θ1 )g(x; q) dx = 0 pj (q) j=1

Dj

⎛ ⎞ r nj ⎜ ⎟ 2 2 ⎝ (x − θ1 ) g(x; q) dx − θ2 g(x; q) dx⎠ = 0 pj (q) j=1

Dj

Dj

which, after rearranging terms and taking θ1 =

nj j

θ22

n

(6.57a)

nj = n into account, become

xg dx g dx

nj (x − θ1 )2 g dx = n g dx

(6.57b)

j

where all integrals are on Dj and, for brevity, we have omitted the functional dependence of g(x; q). For small values of the interval width d and assuming n1 = nr = 0 (i.e. the extreme intervals contain no data) we can ﬁnd an approximate solution of eq. (6.57b) by replacing each function under integral by its corresponding value at the midpoint ξj of the interval Dj . This leads

258 Mathematical statistics to the ‘grouped’ estimate qˆ g = (θˆ1 , θˆ22 ), where 1 θˆ1 ∼ nj ξj = n r−1 j=2

1 θˆ22 ∼ nj (ξj − θˆ1 )2 = n r−1

(6.58)

j=2

In general, the approximate estimates (6.58) are sufﬁciently good for practical purposes even if the extreme intervals are not empty but contain a small part of the data. So, if y = (y1 , . . . , yn ) is the set of data obtained by the experiment (we call the data yi , and not xi as usual, to avoid confusion with the intervals extreme points deﬁned above) we reject the null hypothesis of normality at the level α if t(qˆ g ) =

[nj − npj (qˆ g )] 2 ≥ χα; r−3 npj (qˆ g )

(6.59)

j

2 2 where χ α; r−3 is the α-upper quantile of the distribution χ (r − 3).

As a numerical example, suppose we have n = 1000 observations of a r.v. which we suspect to be normal; also, let the minimum and maximum observed values be ymin = 36.4 and ymax = 98.3, respectively. Let us choose a partition of the real line in r = 9 intervals with d = 10 and D1 = (−∞, 35], D2 = (35, 45] etc. up to D8 = (95, 105] and D9 = (105, ∞). With this partition, suppose further that our data give the absolute frequencies n = (n1 , . . . , n9 ) = (0, 5, 60, 233, 393, 254, 49, 6, 0)

(6.60)

from which, since the intervals midpoints are ξ2 = 40, ξ3 = 50, . . . , ξ8 = 100, we calculate (eq. (6.58)) the grouped estimates m ˆ = 70.02 and sˆ2 = 102.20 for the mean and the variance, respectively. A normal distribution with this mean and variance leads to the (approximate) set of theoretical frequencies np = n(p2 , . . . , p8 ) ∼ = (4.8, 55.5, 241.5, 394.6, 242.4, 56.0, 4.9)

(6.61)

where the approximation lies in the fact that we calculated the integrals (6.56) by using, once again, the value of the function at the midpoints ξj . Finally, using the experimental and theoretical values of eqs. (6.60) 2 and (6.61) we get t(qˆ g ) = 2.36 which is less than χ 0.05; 6 = 12.59 therefore implying that we accept the null hypothesis of normality.

The test of statistical hypotheses

259

This example is just to illustrate the method and in a real case we should use a ﬁner partition, say r ∼ = 15. In this case we would probably have to pool the extreme intervals to comply with the suggestion npj ≥ 5. So, for instance if we choose d = 5 and the ﬁrst two non-empty intervals have the observed frequencies 1 and 4, we can pool these two intervals to obtain an interval of width d = 10 with a frequency count of 5. In this case Cramer [4] suggests to calculate the estimates with the original grouping (i.e. before pooling), and then perform the test by using the quantiles of the distribution χ 2 (r − 3), where r is the number of intervals after pooling. 6.4.2

Kolmogorov–Smirnov test and some remarks on the empirical distribution function

Similarly to the χ 2 goodness-of-ﬁt test, the Kolmogorov–Smirnov test (KS test) concerns the agreement between an empirical distribution and an assumed theoretical one when this latter is continuous. The test is performed by using the PDFs rather than – as the χ 2 does – the pdfs. The relevant statistic is Dn =

sup

−∞<x

This book is essential reading for practising engineers who need a sound background knowledge of probabilistic and statistical concepts and methods of analysis for their everyday work. It is also a useful guide for graduate engineering students and researchers. The theoretical aspects of modern probability theory is the subject of the ﬁrst part of the book. In the second part, it is shown how these concepts relate to the more practical aspects of statistical analyses. Although separated, the two parts of the book are presented in a uniﬁed style and form a wellstructured unity. Moreover, besides discussing a number of fundamental ideas in detail, the author’s approach to the subject matter is particularly useful for the interested reader who wishes to pursue the study of more advanced topics. This book has an unusual combination of topics based on the author’s education as a Nuclear Physicist and his many years of professional activity as a consultant in different ﬁelds of Engineering. Paolo L. Gatti formerly Head of the Vibration Testing and Data Acquisition Division of Tecniter s.r.l., Cassina de’ Pecchi, Milan, Italy, works now as an independent consultant in the ﬁelds of Engineering Vibrations, Statistical Data Analysis and Data Acquisition Systems.

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Probability Theory and Mathematical Statistics for Engineers

Paolo L. Gatti

First published 2005 by Spon Press 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Simultaneously published in the USA and Canada by Spon Press 270, Madison Ave, New York, NY 10016 Spon Press is an imprint of the Taylor & Francis Group © 2005 Paolo L. Gatti This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data Gatti, Paolo L., 1959– Probability theory and mathematical statistics for engineers / Paolo L. Gatti. – 1st ed. p. cm. – (Structural engineering: mechanics and design) Includes bibliographical references and index. 1. Engineering – Statistical methods. 2. Probabilities. I. Title. II. Series. TA240.G38 2004 620 .001 5192–dc22

ISBN 0-203-30543-4 Master e-book ISBN

ISBN 0-203-34366-2 (Adobe eReader Format) ISBN 0-415-25172-9 (Print Edition)

2004006776

To the dearest friends of many years, no matter how far they may be. To my parents Paola e Remo and in loving memory of my grandmother Maria Margherita. Paolo L. Gatti

Contents

Preface

x

PART I

Probability theory

1

1

3

The concept of probability 1.1 1.2 1.3 1.4 1.5

2

Different approaches to the idea of probability 3 The classical deﬁnition 4 The relative frequency approach to probability 14 The subjective viewpoint 18 Summary 19

Probability: the axiomatic approach

22

2.1 2.2 2.3 2.4

Introduction 22 Probability spaces 22 Random variables and distribution functions 33 Characteristic and moment-generating functions 55 2.5 Miscellaneous complements 63 2.6 Summary and comments 72 3

The multivariate case: random vectors 3.1 3.2 3.3

Introduction 76 Random vectors and their distribution functions 76 Moments and characteristic functions of random vectors 85 3.4 More on conditioned random variables 103

76

viii

Contents 3.5 3.6

4

Functions of random vectors 115 Summary and comments 126

Convergences, limit theorems and the law of large numbers 4.1 4.2 4.3 4.4 4.5 4.6 4.7

129

Introduction 129 Weak convergence 130 Other types of convergence 137 The weak law of large numbers (WLLN) 142 The strong law of large numbers (SLLN) 146 The central limit theorem 149 Summary and comments 157

PART II

Mathematical statistics

161

5

163

Statistics: preliminary ideas and basic notions 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

6

Introduction 163 The statistical model and some notes on sampling 164 Sample characteristics 168 Point estimation 180 Maximum likelihood estimates and some remarks on other estimation methods 196 Interval estimation 204 A few notes on other types of statistical intervals 217 Summary and comments 218

The test of statistical hypotheses

223

6.1 6.2 6.3 6.4

Introduction 223 General principles of hypotheses testing 223 Parametric hypotheses 226 Testing the type of distribution (goodness-of-ﬁt tests) 251 6.5 Miscellaneous complements 262 6.6 Summary and comments 271 7

Regression, correlation and the method of least squares 7.1 7.2

Introduction 274 The general linear regression problem 275

274

Contents 7.3 7.4 7.5

Normal regression 285 Final remarks on regression 298 Summary and comments 303

Appendix A: elements of set theory A.1 A.2 A.3

B.3 B.4 B.5

306

Basic deﬁnitions and properties 306 Functions and sets, equivalent sets and cardinality 312 Systems of sets: algebras and σ -algebras 315

Appendix B: the Lebesgue integral – an overview B.1 B.2

ix

318

Introductory remarks 318 Measure spaces and the Lebesgue measure on the real line 319 Measurable functions and their properties 321 The abstract Lebesgue integral 324 Further results in integration and measure theory and their relation to probability 330

Appendix C

337

C.1 The Gamma Function (x) 337 C.2 Gamma distribution 338 C.3 The χ 2 distribution 339 C.4 Student’s distribution 340 C.5 Fisher’s distribution 342 C.6 Some other probability distributions 344 C.7 A few ﬁnal results 350 Index

353

Preface

Tanta animorum imbecillitas est, ubi ratio decessit L.A. Seneca (De Constantia Sapientis, XVII)

A common attitude among engineers and physicists is, loosely speaking, to consider statistics as a tool in their toolbox. They know it is ‘there’, they are well aware of the fact that it can be of great help in a number of cases and, having a general idea of how it works, they ‘dust it off’ and use it whenever the problem under study requires it. A minor disadvantage of this pragmatic, and in many ways justiﬁable (after all, statistics is not their main ﬁeld of study) point of view is that one often fails to fully appreciate its potential, its richness and the complexity of some of its developments. A more serious disadvantage is the risk of improper use, although it is fair to say that this is rarely the case in science when compared with other ﬁelds of activity such as, for instance, politics, advertising and journalism (even assuming the good faith of the individuals involved). These general considerations aside, in the author’s mind the typical reader of this book (whatever the term ‘typical reader’ may mean) is an engineer or physicist who has a particular curiosity – personal and/or professional – for probability and statistics. Although this typical reader is surely interested in statistical techniques and methods of practical use, his/her focus is on ‘understanding’ rather than ‘information’ on this or that speciﬁc method, and in this light he/she is willing to tackle some mathematical difﬁculties in order to, hopefully, achieve this understanding. Since I found myself in this same situation a few years ago and it is now my opinion that the reward is well-worth the effort (this, however, in no way implies that I have reached a full understanding of the subject-matter – unfortunately, I feel that it is not so – it simply means that after a few years of study the general picture is much clearer and many details are much sharper now than then), I decided to write a book which would have fulﬁlled my needs at that time. It goes without saying that there are many good books on the subject – a good number of them are on my shelf and I have often referred to them either for work or in writing this book – and this is why I included a rather detailed list of references at the end of each chapter.

Preface xi The book is divided into two main parts: Part 1 (Chapters 1–4) on probability theory and Part 2 (Chapters 5–7) on mathematical statistics. In addition, three appendices (A, B and C) complement the book with some extra material relevant to the ideas and concepts presented in the main text. With regard to Part 1 on probability, some mathematical difﬁculties arise from the circumstance that the reader may not be familiar with measure theory and Lebesgue integration, but I believe that it would have been unfair to the ‘typical reader’ to pretend to ignore that modern probability theory relies heavily on this branch of mathematics. In Part 2, on the other hand, I tried as much as possible to show the way in which, in essence, this part is a logical – even if more application-oriented – continuation of the ﬁrst; a fact that, although obvious in general, is sometimes not clear in its details. In all, my main goal has been to give a uniﬁed treatment in the hope of providing the reader with a clear wide-angle picture where, in addition, some important details are in good focus. On this basis, in fact, he/she will be able to pursue the study of more advanced topics and understand the main ideas behind the speciﬁc statistical techniques and methods – some of which are rather sophisticated indeed – that he/she will encounter in this and/or other texts. Also, it is evident that in writing a book like this some compromise must be made on the level of mathematical exposition and selection of topics. In regard to the former I have striven for clarity rather than mathematical rigor; in fact, there exist many excellent books written by mathematicians (some of them are included in the references) where rigor is paramount and all the necessary proofs are given in detail. For the latter, it is only fair to say that many important topics, including probably at least one of everyone’s favourites, have been omitted. Out of necessity, in fact, some choices had to be made (a few of them have been made painfully along the way, leaving some doubts that still surface now and then) and I tried to do so with the intention of writing a not-too-long book without sacriﬁcing the spirit of the original idea that had me started in the ﬁrst place. Only the readers will be able to tell if I have been successful and faithful to this idea. Finally, it is possible that, despite the attention paid to reviewing all the material, this book will contain errors, omissions, oversights or misprints. I will be grateful to the readers who spot any of the above or who have any comments for improving the book. Any suggestion will be received and considered. Paolo L. Gatti Milano September 2004

Part I

Probability theory

1

The concept of probability

1.1

Different approaches to the idea of probability

Probabilistic concepts, directly or indirectly, pervade many aspects of human activities, from everyday situations to more advanced and speciﬁc applications in natural sciences, engineering, economy and politics. It is the scope of this introductory chapter to discuss the fundamental idea of probability which, as we will see, is not so obvious and straightforward as it may seem. In fact – in order to deal with practical problems in a ﬁrst stage and to arrive at a sound mathematical theory later – this concept has evolved through the centuries, changing the theory of probability from an almost esoteric discipline to a well-established branch of mathematics. From a strict historical point of view, despite the fact that some general notions have been common knowledge long before the seventeenth century (e.g. Cardano’s treatise ‘Libel de Ludo Aleæ’ (Book of Dice Games) was published in 1663 but written more than a century earlier), the ofﬁcial birth of the theory dates back to the middle of the seventeenth century and its early developments owe much to great scientists such as Pascal (1623–1662), Fermat (1601–1665), Huygens (1629–1695), J. Bernoulli (1654–1705), de Moivre (1667–1754), Laplace (1749–1827) and Gauss (1777–1855). Broadly speaking, probability is a loosely deﬁned term employed in everyday conversation to indicate the measure of one’s belief in the occurrence of a future event when this event may or may not occur. Moreover, we use this word by indirectly making some common assumptions: probabilities near 1 (100%) indicate that the event is extremely likely to occur, probabilities near zero indicate that the event is almost not likely to occur and probabilities near 0.5 (50%) indicate a ‘fair chance’, that is, that the event is just as likely to occur as not. If we try to be more speciﬁc, we can consider the way in which we assign probabilities to events and note that three main approaches have developed through the centuries. Following the common terminology, we call them (1) the classical approach, (2) the relative frequency approach, (3) the subjective approach.

4

Probability theory

This order agrees with the historical sequence of facts. In fact, the classical deﬁnition of probability was the ﬁrst to be given, followed by the relative frequency deﬁnition and – not long before Kolmogorov’s axiomatic approach was introduced in 1931 – by the subjective deﬁnition. Let us examine them more closely.

1.2

The classical deﬁnition

The ﬁrst two viewpoints mentioned in Section 1.1, namely the classical and the relative frequency approaches, date back to a few centuries ago and originate from practical problems such as games of chance and life insurance policies, respectively. Let us consider the classical approach ﬁrst. In a typical gambling scheme, the game is set up so that there exists a number of possible outcomes which are mutually exclusive and equally likely and the gambler bets against the House on the realization of one of these outcomes. The tossing of a balanced coin is the simplest example: there are two equally likely possible outcomes, head or tail, which are mutually exclusive (that is both faces cannot turn up simultaneously) and the bet is, say, the appearance of a head. More speciﬁcally, the classical (or the gambler’s) deﬁnition of probability can be used whenever it can be reasonably assumed that the possible outcomes of the ‘experiment’ are mutually exclusive and equally likely so that one calculates the probability of a particular outcome A as P(A) =

n(A) n(S)

(1.1)

where n(A) is the number of ways in which outcome A can occur and n(S) is the total number of ways in which the experiment can proceed. Note that with this deﬁnition we do not need to actually perform the experiment because eq. (1.1) deﬁnes an ‘a priori’ probability. In tossing a fair coin, for instance, this means that without even trying we can say that n(S) = 2 (head or tail) and the probability of a head is P(A) ≡ P(head) = 1/2. Also, in rolling a fair die – where six outcomes are possible, that is, n(S) = 6 – the appearance of any one particular number can be calculated by means of eq. (1.1) and gives 1/6 while, on the other hand, the appearance of, say, an even number is 1/2. 1.2.1

Properties of probability on the basis of the classical deﬁnition

In the light of the simple examples given in Section 1.2, the classical deﬁnition can be taken as a starting point to give some initial deﬁnitions and determine a number of properties which we expect a ‘probability function’ to have. This will be of great help in organizing some intuitive notions in a more systematic

The concept of probability

5

manner – although, for the moment, in a rather informal way and on the basis of heuristic considerations (the term ‘probability function’ itself is here used informally just to point out that a probability is something that assigns a real number to each possible outcome of an experiment). In order to do so we must turn to the mathematical theory of sets (the reader may refer to Appendix A for some basic aspects of this theory). First of all, we give some deﬁnitions: (a) we call event a possible outcome of a given experiment; (b) among events, we distinguish between simple events, which can happen only in one way, are mutually exclusive and equally likely; (c) compound events, which can happen in more than one way. Then, (d) we call sample space (or event space) the set of all possible simple events. Note that this deﬁnition justiﬁes the fact that simple events are also often called sample points. In the die-rolling experiment, for example, the sample space is the set {1, 2, 3, 4, 5, 6}, a simple event is the observation of a six and a compound event is the observation of an even number (2, 4, or 6). Adopting the notations of set theory, we can view the sample space as a set W whose elements Ej are the sample points. Then, any compound event A is a subset of W and can be viewed as a collection of two or more sample points, that is, as the union of two or more simple events. In the die-rolling experiment above, for example, we can write A = E2 ∪ E4 ∪ E6

(1.2)

where we called A the event ‘observation of an even number’, E2 the sample point ‘observation of a 2’ and so on. In this case, it is evident that P(E2 ) = P(E4 ) = P(E6 ) = 1/6 and, since E2 , E4 and E6 are mutually exclusive we expect an ‘additivity property’ of the form P(A) = P(E2 ∪ E4 ∪ E6 ) = P(E2 ) + P(E4 ) + P(E6 ) = 1/2

(1.3a)

An immediate consequence of eq. (1.3a) is that ⎛ P(W) = P ⎝

6

j=1

⎞ Ej ⎠ =

6

P(Ej ) = 1

(1.3b)

j=1

because it is clear that one of the six faces must necessarily show up. Moreover, if we denote by AC the complement of set A (clearly W = A ∪ AC : for example, in the die experiment if A is the appearance of an even

6

Probability theory

number then the event AC represents the non-occurrence of A, that is, the appearance of an odd number; therefore AC = E1 ∪ E3 ∪ E5 ), we have P(AC ) = 1 − P(A)

(1.4)

If, on the other hand, we consider two events, say B and C, which are not mutually exclusive, a little thought leads to P(B ∪ C) = P(B) + P(C) − P(B ∩ C)

(1.5a)

where P(B ∩ C) is called the compound probability of events B and C, that is, the probability that B and C occur simultaneously. An example will help clarify this idea: returning to our die-rolling experiment, let, for example, B = E2 ∪ E3 and C = E1 ∪ E3 ∪ E6 , then B ∩ C = E3 and, as expected, P(B ∪ C) = (2/6) + (3/6) − (1/6) = (4/6). For three non-mutually exclusive events, say B, C and D, eq. (1.5a) becomes P(B ∪ C ∪ D) = P(B) + P(C) + P(D) − P(B ∩ C) − P(B ∩ D) − P(C ∩ D) + P(B ∩ C ∩ D)

(1.5b)

as the reader is invited to verify. In general, the extension of eq. (1.5a) to n events A1 , A2 , . . . , An leads to the rather cumbersome expression ⎛ ⎞ n n P⎝ Ak ⎠ = P(Ak ) − P(Ak1 ∩ Ak2 ) + · · · + (−1)m+1 k=1

k=1

×

k1 0) are two real parameters (whose meaning is probably well known to the reader but will be shown later). The fact that the pdf (2.29a) satisﬁes the normalization condition (eq. (2.25b)) can be veriﬁed by writing 1 √ σ 2π

+∞ +∞ 1 (t − x)2 2 dt = √ exp − e−(y /2) dy 2σ 2 2π

−∞

−∞

where the second integral is obtained by the change of variable ¯ . +∞ √ y = (x− x)/σ Since from integrals tables we get −∞ exp(−ax2 ) dx = π/a, eq. (2.25b) follows. The PDF of eq. (2.29a) cannot be written in explicit analytical form but it is given by

1 FX (x) = √ σ 2π

x −∞

(t − x) ¯ 2 exp − 2σ 2

1 dt = √ 2π

(x− ¯ ) x/σ

e−(y

2

/2)

dy

−∞

(2.29b) where, again, the second integral is obtained by the same change of variable as above and – due to its importance in statistics – can be easily found in numerical table form. However, if tables are not available, the approximation 1 F(z) ≡ 2π

z −∞

exp(−y2 /2) dy ∼ =

1 1 + exp{−az(1 + bz2 )}

(2.29c)

with a = 1.5976 and b = 0.044715 is sufﬁciently accurate for most applications (the maximum absolute error is ≤ 2 × 10−4 ). Other examples of absolutely continuous probability laws will be given later. As remarked above, in almost all practical cases the continuous singular part of the decomposition is generally absent. As a consequence, it is customary to speak of mixed random variables when neither the absolutely continuous part nor the discrete part of the PDF function are identically zero.

Probability: the axiomatic approach

45

In this case there exist a number of points xn for which eq. (2.17) holds; however eq. (2.18) is no longer true and we have

pn =

n

P{X −1 (xn )} < 1

(2.30)

n

This means that there exist at least a pair of neighboring points xn and xn+1 such that F(xn ) < F(xn+1 −). In other words, F can be written as the sum (called a ‘convex’ linear combination) F(x) = αFac (x) + (1 − α)Fd (x)

(2.31)

where 0 ≤ α ≤ 1, Fac is an absolutely continuous, monotonically increasing function and Fd is a jump function of the type (2.20). Obviously, α = 1 corresponds to the absolutely continuous case and α = 0 to the discrete case. In both cases – and, clearly, also in the mixed case – we will see in the next section how the Lebesgue–Stieltjes integral is the appropriate tool used to calculate important quantities such as the mean value, the variance and, in general, many other parameters which describe in numerical form the behavior of a random variable. Example 2.5

Suppose a r.v. has the following PDF

⎧ ⎪ xb

called the uniform PDF on [a, b]. This function is absolutely continuous and its pdf is f (x) = 1/(b − a) for x ∈ (a, b) and zero elsewhere. Also, it is immediate to determine that a r.v. X whose PDF is given by (2.57a) has moments given by b E(X k ) =

xk f (x) dx = a

bk+1 − ak+1 (b − a)(k + 1)

(2.57b)

Probability: the axiomatic approach

65

from which it follows E(X) = (a + b)/2 and, with a little algebra, Var(X) = (b2 − a2 )/12. The characteristic function is also obtained with little effort and we get ϕ(u) =

eiub − eiua iu(b − a)

(2.57c)

which gives an indeterminate form 0/0 for u = 0; however ϕ(u) → 1 as u → 0. If a = −b the function F(x) is even, the CF is real and can be written as ϕ(u) = sin(bu)/bu. 2.5.2

More on conditional probability

The second aspect we consider here deals with conditional probabilities. We introduced the concept informally in Section 1.2.1 and made some further comments in Section 2.2. There we pointed out that – given a probability space (W, S, P) and a conditioning event G ∈ S with P(G) > 0 – the set function PG : W → [0, 1] deﬁned (for A ∈ S) by the relation PG (A) = P(A ∩ G)/P(G) is a probability function in its own right which, often, is also denoted by P(A|G). Also, if X : W → R is a r.v. on (W, S, P) we observe that X is a r.v. (i.e. measurable) on the space (W, S, PG ) as well, because – we recall from Deﬁnition 2.3 – the measurability of functions is independent on the measure P. We have now two probability measures on (W, S), that is, P and PG , and the ﬁrst thing to note is that PG is absolutely continuous with respect to P because PG (A) = 0 whenever P(A) = 0. Then, the Radon–Nikodym theorem states that there is an essentially unique function H : W → R such that PG (A) =

H dP A

This function is called the Radon–Nikodym derivative of PG with respect to P and it often symbolically denoted by dPG /dP. We state now that H=

IG P(G)

(2.58)

where IG is the indicator function of the set G. In fact, by the deﬁning properties of abstract Lebesgue integral we have P(A ∩ G) =

dP =

A∩G

IA∩G dP =

W

IA IG dP =

W

IG dP A

66 Probability theory where the third equality holds because it is immediate to prove that IA∩G = IA IG . Substituting this result into the deﬁnition of PG we get for every A ∈ S PG (A) =

1 P(G)

IG dP

(2.59)

A

which, in the light of uniqueness of H, proves eq. (2.58). Given a r.v. X on W, a more interesting – and useful in practice – case is when the conditioning set G is the counterimage through X of a Borel set C ⊂ R, that is, when G = X −1 (C). Then, calling PC the probability measure deﬁned by

PC (A) =

P[A ∩ X −1 (C)] P(X −1 (C))

(2.60)

(Incidentally, this notation may seem strange because PC is a measure in (W, S) but C is a Borel set in the domain of X; rigorously one should write PX−1 (C) but then the notation would become too heavy) we can consider its image measure PX|C in R and note that it is absolutely continuous with respect to PX (the image measure of P). By a similar argument as above, the Radon–Nikodym theorem applies. So – recalling the relation between the abstract Lebesgue integrals in dP and dPX and noting that X −1 (B) ∩ X −1 (C) = X −1 (B ∩ C) – from the chain of equalities

PX|C (B) = PC [X

=

−1

1 PX (C)

P[X −1 (B ∩ C] 1 (B)] = = P(X −1 (C)) P(X −1 (C)) dPX =

B∩C

1 PX (C)

dP X −1 (B∩C)

IC dPX

(2.61)

B

it follows that the Radon–Nikodym derivative dPX|C /dPX is dPX|C IC = dPX PX (C)

(2.62)

If now we turn our attention to the conditional PDF deﬁned as FX|C (x) = PC [X −1 ( Jx )]

(2.63)

Probability: the axiomatic approach

67

where Jx = (−∞, x], we can use the basic result of eq. (2.61) to get FX|C (x) =

1 PX (C)

IC dPX = Jx

1 PX (C)

IC dFX

(2.64)

Jx

where the second integral is a Lebesgue–Stieltjes integral. If, in addition, the function FX is absolutely continuous on R then the pdf fX exists and we can take the derivative of both sides to obtain the conditional pdf in terms of the unconditional one fX|C (x) =

IC fX (x) IC fX (x) = PX (C) fX (x) dx

(2.65a)

C

On the other hand, if FX is discrete we have pX|C (xk ) =

IC pX (xk ) IC pX (xk ) = PX (C) xi ∈C p(xi )

(2.65b)

Equations (2.65a) and (2.65b) show that fX|C (or pX|C ) is zero outside the set C while in C, besides the multiplicative constant 1/PX (C), coincides with the unconditioned pdf (or pmf). The factor 1/PX (C) is necessary in order to satisfy the normalization condition. Using the conditional characteristics we can deﬁne and calculate the conditional moments just as we did in the unconditioned case. So, we can deﬁne E(X k |C) =

X k dPC =

xk dPX|C =

R

W

xk dFX|C

(2.66a)

R

where the second equality holds because of the relation between PC and its image measure PX|C and the third equality holds because of the deﬁnition of Lebesgue–Stieltjes integral. Then, by virtue of eq. (2.62) we also have E(X k |C) =

1 PX (C)

xk dFX

(2.66b)

C

where the integral on the r.h.s. is a sum or a Lebesgue integral on R (which in most practical cases coincides with an ordinary Riemann integral) depending on whether FX is discrete or absolutely continuous. Similarly, we can deﬁne the conditional-CF as ϕX|C (u) = E(eiuX |C) and, more generally, the

68 Probability theory expectation of a (Borel) function g(X). In the absolutely continuous case, for example, we get E[g(X)|C] =

Cg(x)fX (x) dx

(2.67)

C fX (x) dx

Clearly, when all events B ⊂ R are independent of the conditioning event C then the conditioned characteristics coincide with the unconditioned ones; the simplest case of this situation is when C = R or, in the space (W, S), when G = W. We close this section with a ﬁnal result regarding conditional expectations which is somehow a counterpart of the total probability formula of eq. (1.12). This result is one form of the so-called total expectation theorem and is given in the following proposition: Proposition 2.27 (Total expectation theorem) Let the sets Gi ∈ S be such that W = ∪i Gi , Gi ∩Gj = ∅ for i = j and P(Gi ) > 0 for all i = 1, 2, . . . . Then E(X) =

P(Gi )E(X|Gi )

(2.68)

i

In fact, as a consequence of the Radon–Nikodym theorem, we can write E(X|Gi ) = [P(Gi )]−1 Gi X dP but then, owing to the properties of the abstract Lebesgue integral we get

X dP =

E(X) = W

X dP =

! i

Gi

X dP

i G i

so that Proposition 2.26 follows from these two results. For the moment, the considerations above sufﬁce and we leave further developments on conditional probability to future sections. In particular, for continuous random variables we will show how one can condition on an event with zero probability, that is an event of the form X = x0 where x0 is a speciﬁed value and PX {x0 } = P(X −1 {x0 }) = 0. 2.5.3

Functions of random variables

The third topic we consider here deals with random variables with a known functional dependence on another random variable. So, let X be a r.v. with known probability distribution FX and let g : R → R be a well-behaved Borel function. Since we already know that the function Y(w) ≡ g(X(w)) : W → R is itself a random variable, we may ask for its probability distribution.

Probability: the axiomatic approach

69

This is not always simple and we will consider only some frequently encountered cases. Suppose ﬁrst that X is absolutely continuous with pdf fX and g is monotonically increasing. Then, given a value y, we have that Y ≤ y whenever X ≤ x (where y = g(x)) and Y is also absolutely continuous. Moreover, in this case the inverse function g −1 exists is single-valued and g −1 (y) = x; therefore FY (y) = P(Y ≤ y) = P[X ≤ g −1 (y)] = FX (g −1 (y))

(2.69a)

then, taking the derivative with respect to y we obtain the pdf of Y as fY (y) = fX (g −1 (y))

dg −1 (y) dy

(2.69b)

If, on the other hand, g is monotonically decreasing, then we have FY (y) = P(Y ≤ y) = P[X > g −1 (y)] = 1 − P[X ≤ g −1 (y)] = 1 − FX (g −1 (y))

(2.70a)

and differentiating fY (y) = −fX (g −1 (y))

dg −1 (y) dy

(2.70b)

By noting that the derivative dg −1 /dy is positive when g is monotonically increasing and negative when g is monotonically decreasing, eqs (2.69b) and (2.70b) can be combined into the single equation fY (y) = fX (g

−1

dg −1 (y) (y)) dy

(2.71)

Example√2.10(a) Let X be a r.v. with pdf fX = e−x (x ≥ 0) and let Y = 2 g(X) = X. Then x = g −1 (y) = y2 , fX (g −1 (y)) = e−y and dg −1 /dy = 2y, 2 so that, by eq. (2.69b), we get fY (y) = 2ye−y (y ≥ 0). Clearly, if we note 2 that FX (x) = 1 − e−x we can use eq. (2.69a) to get the PDF FY (y) = 1 − e−y , as expected, can also be obtained by computing the integral FY (y) = which, y f (t)dt. As an easy exercise, the reader is invited to sketch a graph of fX Y 0 and fY and note that they are markedly different. Example 2.10(b) Let us now consider the linear case Y = g(X) = aX + b. This is an increasing function for a > 0 and decreasing for a < 0. If fX (x)

70 Probability theory is the pdf of X, eq. (2.71) yields fY (y) = |a|−1 fX ((y − b)/a). Also, FY (y) = FX ((y − b)/a) if a > 0 and FY (y) = 1 − FX ((y − b)/a) if a < 0. So, for example, if the original r.v. X is normally distributed – that is, its pdf is given by eq. (2.29a) – and a > 0 we get

(y − ax¯ − b)2 exp − fY (y) = 2a2 σ 2 aσ 2π 1 √

meaning that Y is a Gaussian r.v. itself with mean y¯ = ax¯ + b and variance σY2 = a2 σ 2 . Example 2.10(c) Starting again from the normal pdf of eq (2.29a) we can obtain the pdf of the standardized normal r.v. Y = (X − x)/σ ¯ . Noting that g −1 (y) = σ y + x and dg −1 /dy = σ we get 1 exp(−y2 /2) fY (y) = √ 2π which, as mentioned at the end of Example 2.8, is a normal r.v. with x¯ = 0 and σ = 1. If g is not monotone, it can often be divided into monotone parts; the considerations above then apply to each part and in the end the sum of the various parts is taken. A simple example of this latter case is Y = g(X) = X 2 which is decreasing for x < 0 and increasing for x > 0. Since g(x) is always positive for all x (or, stated differently, g −1 (y) = ∅ for y < 0) the r.v. Y cannot take on negative values and therefore fY (y) = 0 for y < 0. For y > 0 it is left to the reader to determine that the sum of the two parts leads to √ √ √ fY (y) = (2 y)−1 (fX (− y) + fX ( y)). Example 2.11 In applications it is often of interest to have a probabilistic description of the maximum or minimum of a number n of r.v.s. As we will see in later chapters, an important case is when the r.v.s X1 , X2 , . . . , Xn are independent and have the same PDF F(x). Now, ﬁrst of all it can be shown that the function Y = max{X1 , . . . , Xn } is itself a r.v. (as is the minimum). Then, since FY (y) = P(max{X1 , . . . , Xn } ≤ y) = P(X1 ≤ y, . . . , Xn ≤ y) the assumption of independence leads to FY (y) = P(X1 ≤ y, . . . , Xn ≤ y) =

n

i=1

P(X1 ≤ y) = (F(y))n

(2.72a)

Probability: the axiomatic approach

71

and therefore, if F is absolutely continuous fY (y) = n(F(y))n−1 f (y)

(2.72b)

where f is the derivative of F. We note here that the expression P(X1 ≤ y, . . . , Xn ≤ y) is written in the usual ‘shorthand’ notation of probability theory; in rigorous mathematical symbolism, however, this probability is written P ∩ni=1 Xi−1 (−∞, y] . The rigorous notation is useful when we consider the minimum of the r.v.s X1 , X2 , . . . , Xn . In fact, if Jy = (−∞, y] we have −1 FY (y) = P(min{X1 , . . . , Xn }) = P Xi ( Jy ) "

(X −1 ( Jy ))C =P

#C

i

=1−P

i

"

# (X

−1

( Jy ))

C

i

where in the third equality we used de Morgan’s law. Then, by virtue of $ % & $ %C independence P ∩i (Xi−1 ( Jy ))C = i P Xi−1 ( Jy ) = (1 − F(y))n , so that putting the pieces together we ﬁnally get FY (y) = 1 − (1 − F(y))n

(2.73a)

and if F is absolutely continuous (F = f ) fY (y) = n(1 − F(y))n−1 f (y)

(2.73b)

So, for instance, if F is the uniform PDF (eq. (2.57a)) on the interval [a, b] = [0, 1] then FY (y) = n(1 − y)n−1 where 0 ≤ y ≤ 1. If X is a discrete r.v. whose range is the set AX = {x1 , x2 , . . .} = {xi } then Y = g(X) is also a discrete r.v. because its range is the set AY = {y1 , y2 , . . .} = {yk } (note that the elements of AX and AY are labelled by different indexes because, in general, a given value yk may be the image – through g – of more than one xi ). In this case, in general, it is not convenient to go through the PDF but it is better to determine the mass distribution pY by ﬁrst identifying the values yk and then using the relation pY (yk ) = P(Y = yk ) = P(X = g −1 (yk )) =

pX (g −1 (yk ))

(2.74)

xi

where the sum is taken on all values xi (when there is more than one) which are mapped in yk . So, for instance, let X be such that AX = {x1 = −1,

72 Probability theory x2 = 0, x3 = 1}, pX (−1) = pX (0) = 0.25 and pX (1) = 0.5 and let Y = X 2 . Then AY = {y1 , y2 } = {0, 1} and g −1 (y2 ) = −1 ∪ 1 = x1 ∪ x3 so that in calculating pY (y2 ) we must sum the probabilities pX (x1 ) and pX (x3 ). Therefore pY (y2 ) = 0.75. By contrast, the sum is not needed in calculating pY (y1 ) = 0.25.

2.6

Summary and comments

This chapter introduces the axiomatic approach to probability by giving a number of fundamental concepts and results which are at the basis of all further developments in both ﬁelds of probability theory and statistics. In essence, the axiomatic approach consists in calling ‘probability’ any set function that satisﬁes certain properties, together with the deﬁnition of what exactly is meant by the term ‘event’. Clearly, in order to speak of probability this latter deﬁnition is a necessary prerequisite because probabilities can only be assigned to events. Both deﬁnitions, probability and events, are given in Section 2.2 by introducing the concepts of elementary probability spaces – which apply to all cases with a ﬁnite number of possible outcomes – and probability spaces, where the restriction of ﬁniteness is relaxed. These notions are sufﬁciently general to include as special cases all the deﬁnitions of probability considered in Chapter 1. In mathematical terms, a probability space is just a ﬁnite measure space and a probability P is a σ -additive measure deﬁned on a σ -algebra of subsets (the events) of a ‘universal’ set W with the property P(W) = 1. The domain of P is taken to be a σ -algebra because measure theory – by virtue of the construction of the Lebesgue extension of measures – guarantees that a knowledge of the values taken on by P on a limited number of ‘elementary’ events (which, in general, form a semialgebra of subsets of W) is sufﬁcient in order to determine uniquely P on a much broader class of events, this class being, in fact, a σ -algebra. The extension procedure is outlined in Section 2.2.1 and is summarized by the result known as Caratheodory extension theorem. A fundamental aspect of probability which distinguishes it from measure theory is the notion of independent events. Due to its far-reaching consequences in both the theory and real-word applications of statistics, this concept is discussed in some detail in Section 2.2.2, where it is pointed out that the intuitive idea of independence as the absence of causal relation between two (or more) events is translated into mathematical language by a product rule between the probabilities of the events themselves. At this point we consider the fact that in many applications the analyst is mainly interested in assigning probabilities to numerical quantities associated with the outcomes of an experiment rather than to the outcomes themselves. This task is accomplished by introducing the concept of r.v., that is, a real-valued function deﬁned on W and satisfying a ‘measurability’ condition with respect to the σ -algebra of W and the σ -algebra of Borel sets

Probability: the axiomatic approach

73

of the real line R. The condition is formulated by requiring that the inverse image of any Borel set (in the domain of the random variable) be an element of the σ -algebra of W, thus allowing the possibility of assigning probabilities to subsets of R (in the form of open, closed, semiclosed intervals or of individual real numbers, just to mention the most frequently encountered cases). A r.v. X, in turn, induces a probability measure PX on the real line and therefore a real probability space (R, B, PX ). This space, in general, is all that is needed in applications because, through the measure PX , the analyst can obtain a complete probabilistic description of X by deﬁning the so-called PDF of X, usually denoted by FX (x). Clearly, PX and FX are strictly related; in fact, mathematically speaking, PX is the Lebesgue–Stieltjes measure corresponding FX and FX is the generating function the measure PX (this name comes from the fact that in analysis one usually deﬁnes a Lebesgue–Stieltjes measure by means of its generating function and not the other way around as it is done in probability). With the concept of PDF at our disposal, Section 2.3.1 classiﬁes the various types of random variables according to the continuity properties of their PDFs. A ﬁrst classiﬁcation distinguishes between discrete and continuous r.v.s by also introducing the concept of probability mass distribution for discrete r.v.s. Then, among continuous r.v.s a further classiﬁcation distinguishes between absolutely continuous and singular continuous r.v.s, the distinction being due to the fact that for the former type – by far the most important in applications – it is possible to express their PDF by means of the ordinary Lebesgue integral of an appropriate pdf fX (x) which, in turn, is the derivative of FX (x). This possibility relies ultimately on an important result of analysis (given in Appendix B) known as Radon–Nikodym theorem. The conclusion is that the PDF of the most general type of r.v. can be expressed as the sum of a discrete part, an absolutely continuous part and a continuous singular part which, however, is generally absent in most practical cases. Moreover, for the discrete and the absolutely continuous case, respectively, the mass distribution and the pdf provide a complete probabilistic description of the r.v. under study. Proceeding in the discussion of fundamentals, Section 2.3.2 introduces the most common numerical descriptors of r.v.s – the so-called moments of a r.v. – which are deﬁned by means of abstract Lebesgue integrals on the space W. Special cases of moments – the ﬁrst and second moment, respectively – are the familiar quantities known as mean and variance. The properties of moments are then considered together with the important result of Chebychev’s inequality. Subsequently, the problem of actually calculating moments is considered by ﬁrst noting that it is generally not convenient to compute abstract integrals on W. In this regard, in fact, it is shown that moments can be obtained as Lebesgue–Stieltjes integrals on the real line and that these integrals, in the most common cases of discrete and absolutely continuous r.v.s respectively, reduce to a sum and to an ordinary Riemann

74 Probability theory integral. A few examples are then given in order to illustrate some frequently encountered cases. Besides the PDF, another way of obtaining a complete probabilistic description of a r.v. X is its characteristic function ϕX (u). The concept is introduced in Section 2.4 together with some comments on the ‘parallel’ notion of moment generating function (denoted MX (s)). The main properties of CFs are given by also showing how moments (when they exist) can be easily computed by differentiating ϕX . The fact that the CF provides a complete probabilistic description of a r.v. is due to the existence of a one-to-one relationship between PDFs and CFs. The problem of obtaining the CF from the PDF is given by the deﬁnition of CF itself while the reverse problem is addressed by the so-called inversion formulas which, in the general case, are important for their mere existence but are of little practical use. In the particular case of absolutely continuous r.v.s, however, things are easier because the correspondence reduces to the fact that the pdf fX and the CF ϕX are a Fourier transform pair and the notion of Fourier transform is well known and widely used in Engineering and Physics literature. The section closes with a brief discussion of convergence in distribution (or weak convergence) of sequences of r.v.s for its strict relation with pointwise convergence of characteristic functions. Finally, in Section 2.5 we consider a number of complementary ideas and concepts which are worthy of mention in their own right but have been delayed in order not to interrupt the main line of reasoning. Section 2.5.1 introduces the notion of almost-sure (and almost impossible) events by pointing out that there exist events with probability one (and zero) which are different from W (and ∅). This is not surprising in the light of measure theory when, for example, one considers the Lebesgue measure on a ﬁnite interval of the real line. Subsequently (Section 2.5.2) we extend the notion of conditional probability by showing how, in general – given an event with strictly positive probability – there is no difﬁculty in deﬁning such concepts as the conditional PDF of a r.v., the conditional CF, etc. The chapter closes with Section 2.5.3 where it is shown with some examples how to obtain the PDF, pdf or mass distribution of a function g(X) when the PDF (pdf or mass distribution) of the r.v. X is known.

References and further reading [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press,19th printing (1999). [3] Gnedenko B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [4] Haaser, N.B., Sullivan, J.A., ‘Real Analysis’, Dover, New York (1991). [5] Kolmogorov, A.N., ‘Foundations of Probability’, AMS Chelsea Publishing, Providence, Rhode Island (2000).

Probability: the axiomatic approach

75

[6] Kolmogorov, A.N., Fomin, S.V., ‘Introductory Real Analysis’, Dover, New York (1975). [7] Kolmogorov, A.N., Fomin, S.V., ‘Elementi di Teoria delle Funzioni e di Analisi Funzionale’, Edizioni Mir, Mosca (1980). [8] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [9] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000). [10] Rudin, W., ‘Principles of Mathematical Analysis’, 3rd ed., McGraw-Hill, New York, (1976). [11] Rudin, W., ‘Real and Complex Analysis’, McGraw-Hill, New York (1966). [12] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997).

3

3.1

The multivariate case: random vectors

Introduction

The scope of this chapter is to proceed along the line of reasoning of Chapter 2 by turning our attention to cases in which two or more random variables are considered together. With this in mind, we will introduce the new concept of ‘random vector’ by considering measurable vector-valued functions deﬁned on probability spaces. The main mathematical aspects parallel closely the one-dimensional case but it is worth pointing out that now the notion of stochastic independence will play a major role. In fact, this concept is peculiar to probability theory and distinguishes it from being merely an application of analysis.

3.2

Random vectors and their distribution functions

The deﬁnition of random vector is a straightforward generalization of the concept of random variable; in fact Deﬁnition 3.1 Given a probability space (W, S, P), an n-dimensional random vector is a function X: W → Rn such that X−1 (B) ∈ S for every Borel set B ∈ B(Rn ) where, as customary, we denote by B(Rn ) or Bn the σ -algebra of all Borel sets of Rn . In this regard it is important to note that the σ -algebra Bn is the cartesian product of the n terms B × B × · · · × B, meaning, in other words, that every n-dimensional Borel set A ∈ Bn is of the form A = A1 × A2 × · · · × An where A1 , . . . , An are one-dimensional Borel sets. So, in other words, a random vector is a measurable function from W to Rn just as a random variable is a measurable function from W to the real line R. In the present case, however, the vector-valued function X has n components – that is, X = (X1 , X2 , . . . , Xn ) – and the question on the measurability of each individual function Xi (i = 1, . . . , n) arises. The main result is that X is measurable if and only if each function Xi is measurable,

The multivariate case

77

or, equivalently: Proposition 3.1 The vector-valued function X is a random vector if and only if each one of its components Xi is a random variable. As in the one-dimensional case, the original probability space (W, S, P) is of little importance in applications and one should not worry too much about measurability because the concept is sufﬁciently broad to cover almost all cases of practical interest. Therefore, given a random vector X, the analyst’s main concern is the (real) induced probability space (Rn , Bn , PX ), where the probability measure PX is deﬁned by the relation PX (B) ≡ P[X−1 (B)] = P{w ∈ W : X(w) ∈ B}

(3.1a)

for all B ∈ Bn (it is not difﬁcult to show that PX is, indeed, a probability measure). Again, we note that a common ‘shorthand’ notation is to write P(X ∈ B) to mean the probability deﬁned by eq. (3.1a). Also, in the light of the fact that B can be expressed as the cartesian product of n one-dimensional Borel sets B1 , . . . , Bn , the explicit form of eq. (3.1a) is n −1 −1 −1 Xi (Bi ) PX (B) ≡ P[X (B)] = P[X (B1 × · · · × Bn )] = P i=1

(3.1b) By means of PX we can deﬁne the so-called joint probability distribution function (joint-PDF) FX : Rn → [0, 1] as FX (x) = PX {w ∈ W : X1 (w) ≤ x1 , X2 (w) ≤ x2 , . . . , Xn (w) ≤ xn } (3.2a) where x ∈ Rn is the vector whose components are x1 , x2 , . . . , xn and it is understood that all the inequalities on the r.h.s. of eq. (3.2a) must hold simultaneously. In rigorous (and rather cumbersome) notation it may be worth noting that FX (x) can be expressed in terms of the original probability P as n −1 {Xi (−∞, xi ]} FX (x) = P (3.2b) i=1 (and probably this is why, in agreement with the ‘shorthand’ notation above, one often ﬁnds the less intimidating FX (x) = P(X1 ≤ x1 , X2 ≤ x2 , . . . , Xn ≤ xn )). The main properties of the joint-PDF are the natural extensions of (D1)–(D3) given in Chapter 2 and can be summarized as follows: (D1 ) FX (x) = FX (x1 , x2 , . . . , xn ) is non-decreasing and continuous to the right in each variable xi (i = 1, . . . , n),

78 Probability theory (D2 ) limxi →−∞ FX (X) = 0 and limX→∞ FX (x) = 1, where it should be noted that the ﬁrst limit holds for any particular xi tending to −∞ (with all other coordinates ﬁxed) whereas the second property requires that all xi tend to +∞. So, in a different notation, the two properties can be expressed as FX (−∞, x2 , . . . , xn ) = FX (x1 , −∞, . . . , xn ) = · · · = FX (x1 , x2 , . . . , −∞) = 0; FX (+∞, +∞, . . . , +∞) = 1 respectively. In mathematical terminology – as in the one-dimensional case (Section 2.3) – one refers to PX as the Lebesgue–Stieltjes measure determined by FX and, conversely, to FX as the generating function of the (ﬁnite) measure PX . If now we turn our attention to the property expressed by eq. (2.15), we ﬁnd that its multi-dimensional generalization is a bit more involved. For simplicity, let us consider the two-dimensional case ﬁrst. The two-dimensional counterpart of the half-open interval (a, b] is a rectangle R whose points satisfy the inequalities a1 < x1 ≤ b1 and a2 < x2 ≤ b2 ; with this in mind it is not difﬁcult to determine that P(X ∈ R) = FX (b1 , b2 ) − FX (b1 , a2 ) − FX (a1 , b2 ) + FX (a1 , a2 )

(3.3a)

and going over to the more complicated n-dimensional case we get P(X ∈ R) =

(−1)k FX (c1 , c2 , . . . , cn )

(3.3b)

where now (i) R is the n-dimensional parallelepiped (a1 , b1 ] × (a2 , b2 ] × · · · × (an , bn ], (ii) the sum is extended to all the 2n possible choices of the ci ’s being equal to ai or bi – that is, the vertexes of the parallelepiped – and (iii) k represents the number of ci ’s being equal to ai . For instance, if n = 3 we get P(X ∈ R) = FX (b1 , b2 , b3 ) − FX (b1 , b2 , a3 ) − FX (b1 , a2 , b3 ) − FX (a1 , b2 , b3 ) + FX (b1 , a2 , a3 ) − FX (a1 , b2 , a3 ) + FX (a1 , a2 , b3 ) − FX (a1 , a2 , a3 ) So, to every random vector there corresponds a joint-PDF which satisﬁes the properties above. The reverse statement, however, is not true in general unless we add another property to (D1 ) and (D2 ): for a function F to be the joint-PDF of some random vector the sum on the r.h.s. of eq. (3.3b) must be non-negative for any ai , bi such that ai ≤ bi (i = 1, 2, . . . , n). If a function F satisﬁes these three properties, then it is the joint-PDF of some random

The multivariate case

79

vector although, as in the one-dimensional case, this vector is not uniquely determined by F. This is only a minor inconvenience without signiﬁcant consequences in most practical cases. If all the components of a random vector are discrete random variables, we speak of discrete random vector. More speciﬁcally, a random vector X = (X1 , X2 , . . . , Xn ) is discrete if there is a ﬁnite or countable set AX ⊂ Rn such that P[(X1 , X2 , . . . , Xn ) ∈ AX ] = 1; in this case – besides being understood that the set AX is the range of X – the function pX : Rn → [0, 1] deﬁned by pX (x1 , x2 , . . . , xn ) ≡ P(X1 = x1 , X2 = x2 , . . . , Xn = xn )

(3.4)

is called the joint probability mass function (joint-pmf) of X and satisﬁes the normalization condition

pX =

all x

···

all x1

pX (x1 , . . . , xn ) = 1

(3.5)

all xn

The other type of random vector commonly encountered in applications is called jointly absolutely continuous. In this case there is a measurable non-negative function fX on Rn such that for all B ∈ Bn we have PX (B) =

fX dµn

(3.6)

B

where µn denotes here the n-dimensional Lebesgue measure. The function fX (x1 , x2 , . . . , xn ) is called the joint probability density function (joint-pdf) of X and its main properties are the generalization of the one-dimensional case (eqs (2.23), (2.25a) and (2.25b)), that is x1 x2 FX (x1 , x2 , . . . , xn ) =

xn ···

−∞ −∞

fX (t1 , t2 , . . . , tn ) dt1 dt2 · · · dtn

−∞

(3.7a) fX (x1 , . . . , xn ) = ∞ ∞

(3.7b)

∞ ···

−∞ −∞

∂FX (x1 , . . . , xn ) ∂x1 · · · ∂xn

fX (x1 , x2 , . . . , xn ) dx1 dx2 · · · dxn = 1

(3.7c)

−∞

As in the one-dimensional case, discrete and absolutely continuous random vectors, or combinations thereof, are not the only possibilities because Proposition 2.11 on the decomposition of measures still holds in Rn and the

80 Probability theory decomposition of a general PDF, in turn, reﬂects the decomposition of its probability measure. The cases shown above, however, are by far the most common in applications and there is generally no need – besides a speciﬁc theoretical interest – to introduce further complications which, if and whenever necessary, will be considered in future discussions. So, we close this section here and turn our attention, once again, to the important role of stochastic independence. 3.2.1

Marginal distribution functions and independent random variables

In the preceding section we pointed out that each individual component Xi of a random vector X = (X1 , . . . , Xn ) is a random variable itself. Consequently, it becomes important to examine the relation between the joint-PDF of X and the PDF of its components in order to answer the following two questions: (i) given the joint-PDF of X is it possible to determine the PDF of each Xi or the joint-PDF of some of the Xi taken together and forming a random vector with m(m < n) components? (ii) given all the PDFs Fi (x) of each Xi is it possible to obtain the joint-PDF FX (x1 , . . . , xn ) of the random vector X? Let us consider question (i) ﬁrst. The answer is always yes because the joint-PDF of X implicitly contains the joint-PDF of any vector obtained by eliminating some of its components. This PDF can be determined from FX by letting all the components to be eliminated tend to inﬁnity; so, if we call Y the vector obtained by eliminating the kth component of X we have FY (x1 , . . . , xk−1 , xk+1 , . . . , xn ) = lim FX (x1 , . . . , xn ) xk →∞

(3.8)

Similarly, if we eliminate any 2, 3, . . . , n − 1 components of X the PDF of the new vector will be a function of the remaining n − 2, n − 3, . . . , 1 variables, respectively, and the r.h.s. of eq. (3.8) will be a multiple limit where all the variables to be eliminated tend to +∞. All the possible ‘sub-PDFs’, so to speak, obtained like this are called marginal-PDFs of the original vector X. Consider the two-dimensional case as an example; here we have a vector X = (X, Y) whose joint-PDF is the function FX (x, y) (often also denoted by FXY (x, y)) and the two marginal-PDFs are the one-dimensional PDFs of the random variables X and Y, respectively, that is, FX (x) = FXY (x, ∞) ≡ lim FXY (x, y) y→∞

FY (y) = FXY (∞, y) ≡ lim FXY (x, y) x→∞

(3.9)

The multivariate case

81

where, clearly, FX (x) is associated to the probability measure PX : B → [0, 1] and FY (y) is associated to the probability measure PY : B → [0, 1] (we recall that B is the collection of all Borel sets of the real line). So, the ﬁrst part of eq. (3.9) tells us that FX (x) is the probability that the r.v. X takes on a value less than or equal to x when all the possible values of Y have been taken into account and, similarly, the second part of eq. (3.9) states that FY (y) is the probability that the r.v. Y takes on a value less than or equal to y when all the possible values of X have been taken into account. Therefore – continuing with the two-dimensional case for simplicity – it is not difﬁcult to see that the marginal-pmfs of a discrete random vector are given by pX (x) =

pXY (x, y)

all y

pY (y) =

(3.10a)

pXY (x, y)

all x

where pXY (x, y) is the joint-pmf of the two r.v.s X, Y forming the vector X. Similarly, if X is a two-dimensional absolutely continuous random vector with joint-pdf fXY (x, y) = ∂FXY /∂x∂y, the two marginal (one-dimensional) pdfs are fX (x) =

+∞ fXY (x, y) dy −∞ +∞

fY (y) =

(3.10b)

fXY (x, y)dx −∞

Example 3.1(a) joint-pmf is

Let X be a discrete two-dimensional random vector whose

pXY (x, y) = (1 − q)2 q x+y

(3.11)

where q is a constant 0 ≤ q < 1 and the variables x, y can only take on natural values (i.e. 0, 1, 2, . . .). Then eq. (3.10a) yields for the marginal-pmfs pX (x) = q x (1 − q)2 pY (y) = qy (1 − q)2

qy = (1 − q)q x

all y q x = (1 − q)qy

(3.12)

all x

because the series n qn converges to (1 − q)−1 whenever 0 ≤ q < 1. In addition, the reader is invited to verify that the joint-pmf (3.11) satisﬁes the

82 Probability theory normalization condition of eq. (3.5) which, in this case, is written

(1 − q)2 q x+y = 1

all x all y

Example 3.1(b) Let the two-dimensional absolutely continuous random vector X have the joint-pdf fX (x, y) =

( ' 1 1 √ exp − (x2 + xy + y2 ) 3 2π 3

(3.13a)

then, using the ﬁrst of eqs (3.10b) we can obtain the marginal-pdf of the r.v. X by integrating (3.13a) in dy over the entire real line. In order to do so we start by rewriting fX (x, y) as

1 x 2 1 x2 fX (x, y) = exp − y+ √ exp − 4 3 2 2π 3

(3.13b)

so that the ﬁrst exponential can be factored out of the integral in dy. Then, by performing the change of variable t = y + x/2 we get 1 fX (x) = √ exp(−x2 /4) 2π 3

∞

−∞

exp(−t 2 /3) dt (3.14a)

1 = √ exp(−x2 /4) 2 π ∞ √ where in the last equality we used the result −∞ exp(−ax2 ) dx = π/a (which can easily be found in integral tables). Finally, by symmetry, it is immediate to obtain 1 fY (y) = √ exp(−y2 /4) 2 π

(3.14b)

If now we consider question (ii) posed at the beginning of this section it turns out that its answer, in the general case, is no. More speciﬁcally, one cannot determine the joint-PDF of the random vector X = (X1 , . . . , Xn ) from the PDFs of its components Xi unless they are independent. In mathematical terms the following proposition holds Proposition 3.2(a) Let X1 , . . . , Xn be random variables on the probability space (W, S, P), let Fi (xi ) be the PDF of Xi (i = 1, . . . , n) and FX (x1 , . . . , xn )

The multivariate case

83

be the joint-PDF of the vector X = (X1 , . . . , Xn ). Then X1 , . . . , Xn are independent if and only if FX (x1 , . . . , xn ) = F1 (x1 )F2 (x2 ) . . . Fn (xn )

(3.15)

for all real x1 , . . . , xn . It should be noted that Proposition 3.2(a) is an ‘if and only if’ statement; this means that if X1 , . . . , Xn are independent then their joint-PDF can be obtained by taking the product of the individual PDFs and, conversely, if the joint-PDF of a random vector X is the product of n one-dimensional PDFs, then the components X1 , . . . , Xn are independent random variables. At this point, however, we must take a step back and return to the notion of stochastic independence introduced in Chapter 2. In Section 2.2.2, in fact, we discussed in some detail the notion of stochastic independence of events but nothing has been said on independent random variables; we do it now by giving the following deﬁnition Deﬁnition 3.2 A collection of random variables X1 , X2 , . . . is called an independent collection if for any arbitrarily chosen class of Borel sets B1 , B2 , . . . the events X1−1 (B1 ), X2−1 (B2 ), . . . are collectively independent. This deﬁnition means that the product rule (2.9) must apply. So, in particular, n random variables X1 , . . . , Xn are called independent if for any choice of Borel sets B1 , . . . , Bn we have ⎛ P⎝

n

k=1

⎞ Xk−1 (Bk )⎠ =

n

P[Xk−1 (Bk )]

k=1

which, in turn, implies that PX (B1 × B2 × · · · × Bn ) = PX1 (B1 )PX2 (B2 ) · · · PXn (Bn ). We have the following result: Proposition 3.2(b) Let X1 , . . . , Xn be random variables on the probability space (W, S, P), then they are independent if and only if the measure PX is the product of the n individual PXi (i = 1, 2, . . . , n). In the light of the fact that the individual PDFs Fi (xi ) are deﬁned by the probabilities PXi , it is not surprising that Proposition 3.2(a), as a matter of fact, is a consequence of Proposition 3.2(b). Also – as it has been done for events – one can introduce the concept of ‘collection of pairwise independent random variables’ – that is, a set of r.v.s X1 , X2 , . . . where Xi is independent of Xj for each pair of distinct indexes i, j – and note that pairwise independence does not imply independence. The converse, however, is true and it is

84 Probability theory evident that these two statements parallel closely the remarks of Chapter 2 (Section 2.2.2). For discrete and absolutely continuous random variables independence can be characterized in terms of pmfs and pdfs, respectively, because the product rule applies to these functions. More speciﬁcally, if X1 , . . . , Xn are a set of independent random variables on a probability space (W, S, P) then, with obvious meaning of the symbols, pX (x1 , . . . , xn ) = p1 (x1 )p2 (x2 ) · · · pn (xn )

(3.16a)

in the discrete case and fX (x1 , . . . , xn ) = f1 (x1 )f2 (x2 ) · · · fn (xn )

(3.16b)

in the absolutely continuous case. Conversely, if – as appropriate – eq. (3.16a) or (3.16b) applies then the random variables X1 , . . . , Xn are independent. In this regard, for example, it may be worth noting that the two random variables X, Y of Example 3.1(a) are independent because (see eqs (3.11) and (3.12)) pXY (x, y) = pX (x)pY (y). On the other hand, the random variables X, Y of Example 3.1(b) are not independent; in fact, in this case the joint-pdf fX (x, y) cannot be factored as required in eq. (3.16b) because of the cross-term xy in the exponential. One word of caution on the absolutely continuous case is in order: if the random vector X has a pdf fX then each Xi has a pdf fi and eq. (3.16b) holds if X1 , . . . , Xn are independent. However, from the fact that each Xi has a density it does not necessarily follow that X = (X1 , . . . , Xn ) has a density; it does if X1 , . . . , Xn are independent and this density fX is given – as shown by eq. (3.16b) – by the product of the n pdfs fi . As a ﬁnal remark for this section we point out an important property of independent random variables: measurable functions of independent r.v.s are independent r.v.s. More speciﬁcally, we can state the following result whose proof, using the deﬁnition of independence of the Xi , is almost immediate. Proposition 3.3 Let X1 , . . . , Xn be a set of independent random variables and g1 , . . . , gn a set of Borel functions. Then the random variables Z1 , . . . , Zn (we recall from Chapter 2 that Borel functions of r.v.s are r.v.s themselves) deﬁned by the relations Zi ≡ gi (Xi ) (i = 1, 2, . . . , n) are independent. More generally, if Y1 , . . . , Ym are sub-vectors of a vector X such that none of the components of X is a component of more that one of the Yj and g1 , . . . , gm are measurable functions, then Zi ≡ gi (Yi )(i = 1, 2, . . . , m) are independent. Also, one can proceed further. In fact, it is possible to extend Deﬁnition 3.1 in order to deﬁne the independence of n random vectors X1 , . . . , Xn and determine that also in this case the factorization property of the PDFs is a necessary and sufﬁcient condition for independence. For the

The multivariate case

85

moment, however, the results given here will sufﬁce and we delay further considerations on independence to later sections.

3.3

Moments and characteristic functions of random vectors

For simplicity, let us ﬁrst consider a two-dimensional random vector X = (X, Y) deﬁned on a probability space (W, S, P). This is a frequently encountered case in applications and it is worthy of consideration in its own right before generalizing to higher dimensional vectors. As in the one-dimensional case (Section 2.3.2), the moments are deﬁned as abstract Lebesgue integrals in dP. So, if i, j are two non-negative integers the joint-moments of order i + j – denoted by E(X i Y j ) or mij – are deﬁned as mij = E(X i Y j ) =

X i Y j dP

(3.17a)

W

which, in the absolutely continuous case, becomes ∞ ∞ xi yj f (x, y) dx dy

mij =

(3.17b)

−∞ −∞

and the integrals are replaced by the appropriate sums in the discrete case. In the light of eq. (3.10b) and their discrete counterparts (3.10a), it is then clear that the ﬁrst-order moments m10 and m01 , respectively, are simply µX = E(X) and µY = E(Y), that is, the mean values of the individual random variables X and Y and, similarly, mi0 and m0j are the ith moment of X and the jth moment of Y. The central (joint) moments of order i +j, in turn, are deﬁned as (provided that µX , µY < ∞) µij = E[(X − µX )i (Y − µY )j ]

(3.18)

where, in the important case i + j = 2 (second-order central moments) we have µ20 = σX2 = Var(X) and µ02 = σY2 = Var(Y). The moment µ11 is given a special name and is called the covariance of the two variables X, Y. For this reason µ11 is often denoted by Cov(X, Y) – although the symbols XY , σXY and KXY are also frequently found in literature. Besides the immediate relations Cov(X, X) = σX2 , Cov(Y, Y) = σY2 and Cov(X, Y) = Cov(Y, X) it should also be noted that the notion of covariance was mentioned in passing

86 Probability theory in Proposition 2.15 (Section 2.3.2) where, in addition, it was shown that Cov(X, Y) = E(XY) − E(X)E(Y) = m11 − µX µY

(3.19a)

This equation is, broadly speaking, the ‘mixed-variables’ counterpart of eq. (2.34), which we rewrite here for the two individual r.v.s X, Y σX2 = µ20 = E(X 2 ) − E2 (X) = m20 − µ2X σY2 = µ02 = E(Y 2 ) − E2 (Y) = m02 − µ2Y

(3.19b)

Some of the main properties of the abstract integral can be immediately be re-expressed in terms of moments. We have already considered linearity for n random variables (Proposition 2.13(c)) – which, in our present case of two r.v.s reads E(aX + bY) = aE(X) + bE(Y) where a, b are any two real or complex constants – but, in particular, we want to point out here the following inequalities (a) Holder’s inequality: let p, q be two numbers such that p > 1, q > 1 and 1/p + 1/q = 1, then E(|XY|) ≤ [E(|X|p )]1/p [E(|Y|q )]1/q

(3.20)

(b) Cauchy–Schwarz inequality ) ) 2 E(|XY|) ≤ E(X ) E(Y 2 )

(3.21)

Both relations are well known to the reader who is familiar with the theory of Lebesgue-integrable function spaces and, clearly, eq. (3.21) is a special case of (3.20) when p = q = 2. In particular – since the Cauchy–Schwarz inequality holds for any two r.v.s – there is no loss of generality in considering the two centered r.v.s W = X − µX , Z = Y − µY and rewriting (3.21) as E(WZ) = Cov(XY) ≤ σW σZ = σX σY , where the last equality holds because the variance of a constant is zero. Therefore, if one deﬁnes the correlation coefﬁcient ρXY as ρXY =

Cov(XY) σX σY

(3.22)

it is immediate to determine that −1 ≤ ρXY ≤ 1. The fact that whenever ρXY = −1 or ρXY = 1 there is a perfect linear relationship between the two r.v.s X and Y (i.e. a relation of the type Y = aX + b, where a, b are two constants) is not so immediate and requires some explanation. In order to do this as an exercise, note that both equalities ρXY = ±1 imply

The multivariate case

87

E2 (WZ)/E(W 2 )E(Z2 )

= 1, where W, Z are the two centered r.v.s deﬁned above. This relation can be rewritten as −

E2 (WZ) = −E(Z2 ) or, equivalently E(W 2 )

E2 (WZ) E(WZ) E(W 2 ) − 2 E(WZ) + E(Z2 ) = 0 E2 (W 2 ) E(W 2 ) so that setting E(WZ)/E(W 2 ) = a we get a2 E(W 2 )−2aE(WZ)+E(Z2 ) = 0, that is E[(aW − Z)2 ] = 0. This, in turn, means that aW − Z = const, that is, that W and Z are linearly related. Then, since (by deﬁnition) there is a linear relation between W and X and between Z and Y, it follows that also X and Y must be linearly related. On the other hand, in order to prove that Y = aX + b implies ρXY = 1 or ρXY = −1 (depending on whether a > 0 or a < 0) it is sufﬁcient to note that in this case Cov(XY) = aσX and σY = |a|σX ; consequently, ρXY = ±1 follows from the deﬁnition of correlation coefﬁcient. The opposite extreme to maximum correlation occurs when ρXY = 0, that is, when Cov(XY) = 0 (if, as always implicitly assumed here, both σX , σY are ﬁnite and different from zero). In this case we say that X and Y are uncorrelated and then, owing to eq. (3.19a), we get E(XY) = E(X)E(Y). This form of ‘multiplication rule’ for expected values may suggest independence of the two random variables because the following proposition holds: Proposition 3.4

If X, Y are two stochastically independent r.v.s then

E(XY) = E(X)E(Y)

(3.23)

and therefore Cov(XY) = 0. This result can be proven by using the factorization properties given in eqs (3.15), (316a) and (3.16b), but the point here is that the reverse statement of Proposition 3.4 is not, in general, true (unless in special cases which will be considered in future sections). In fact, it turns out that uncorrelation – that is, Cov(XY) = 0 – is a necessary but not sufﬁcient condition for stochastic independence. In other words, two uncorrelated r.v.s are not necessarily unrelated (a term which, broadly speaking, is a synonym of independent) because uncorrelation implies a lack of linear relation between them but not necessarily a lack of relation in general. The following example illustrates this situation. Example 3.2 Consider a random vector (X, Y) which is uniformly distributed within a circle of radius r centered about the origin. This means

88 Probability theory that the vector is absolutely continuous with joint-pdf given by 1/π r2 , x2 + y2 < r2 fXY (x, y) = 0, otherwise so that, for instance, if X = 0 then Y can have any value between −r and r but if X = r then Y can only be zero. Therefore, since knowledge of X provides some information on Y, the two variables are not independent. On the other hand, they are uncorrelated because the symmetry of the problem leads to the result Cov(XY) = 0. In fact, denoting by C the domain where fXY = 0, we can calculate the covariance as (see eq. (3.44)) Cov(XY) =

xy f (x, y) dx dy = C

1 π r2

xy dx dy C

and all the integrals in the four quadrants have the same absolute value. However, the function xy under the integral sign is positive in the ﬁrst and third quadrant and negative in the second and fourth quadrant so that summing all the four contributions yields Cov(XY) = 0. As a simpler example consider a r.v. X with the following characteristics: (a) its pdf (or pmf if it is discrete) is symmetrical about the ordinate axis and (b) it has a ﬁnite fourth moment. Then, if we deﬁne the r.v. Y = X 2 it is immediate to determine that X and Y are uncorrelated but not independent. The deﬁnition of characteristic function (or, more precisely, joint-CF) for a two-dimensional random vector is a simple extension of the one-dimensional case of Section 2.4 and we have φ(u, v) = E[ei(uX+vY) ]

(3.24a)

which, in view of generalization to higher dimensions, can be expressed more synthetically with the aid of matrix algebra. We denote by u the vector whose components are the two real variables u, v and write ϕX (u) = E[exp(iuT X)]

(3.24b)

where, following the usual matrix notation, two-dimensional vectors are expressed as column matrices and their transpose (indicated by the upper T) are therefore row matrices. So, in eq. (3.24b) it is understood that the matrix

The multivariate case

89

multiplication in the exponential reads X uT X = (u, v) = uX + vY Y The properties of the CF of a random vector parallel closely the onedimensional case; in particular ϕX (u) is uniformly continuous on R2 and, in addition ϕX (0) = 1 |ϕX (u)| ≤ 1 for all u ∈ R2 ∗ ϕX (−u) = ϕX (u)

(3.25)

where 0 = (0, 0) is the zero vector and the asterisk denotes complex conjugation. Also, if we set n = j + k (where j, k are two integers) and the vector X has ﬁnite moments of order n, then ϕX (u, v) is j times derivable with respect to u and k times derivable with respect to v and ∂ n ϕX (u, v) n n j k i mjk = i E(X Y ) = (3.26) ∂ j u∂ k v u=v=0 which is the two-dimensional counterpart of eq. (2.47a). Equation (3.26) shows that the moments of a random vector coincide – besides the multiplicative factor 1/in – with the coefﬁcients of the MacLaurin expansion of ϕX . This implies that the existence of all moments allows one to construct the MacLaurin series of the CF although, as in the one-dimensional case, in general it does not allow to reconstruct ϕX itself. The marginal CFs of any ‘sub-vector’ of X can be obtained from ϕX by simply setting to zero all the arguments corresponding to the random variable(s) which do not belong to the sub-vector in question. This is an immediate consequence of the deﬁnition of CF and in the two-dimensional case under study we have ϕX (u) = ϕX (u, 0) ϕY (v) = ϕX (0, v)

(3.27)

As ﬁnal remarks to this section, two results are worthy of notice. The ﬁrst is somehow expected and states: Proposition 3.5 Two random variables X, Y forming a vector X are stochastically independent if and only if ϕX (u, v) = ϕX (u)ϕY (v)

(3.28)

meaning that the product rule (3.28) is a necessary and sufﬁcient condition for independence.

90 Probability theory A word of caution is in order here because Proposition 3.5 should not be confused with a different result (see also Example 3.3) which states that if X, Y are independent, then ϕX+Y (u) = ϕX (u)ϕY (u)

(3.29)

In fact, the independence of X and Y imply the independence of the r.v.s ei uX and ei uY and consequently E(ei u (X+Y) ) = E(ei uX ei uY ) = E(ei uX )E(ei uY ) by virtue of Proposition 3.4. Then, by the deﬁnition of CF we get E(ei uX )E(ei uY ) = ϕX (u)ϕY (u). The converse of this result, however, is not true in general and the equality (3.29) does not imply the independence of X and Y. These same considerations, clearly, can be extended to the case of more than two r.v.s. The second remark – here already given in the n-dimensional case – has to do with the important fact that the joint-CF ϕX provides a complete probabilistic description of the random vector X = (X1 , . . . , Xn ) because the joint-PDF FX (x1 , . . . , xn ) is uniquely determined by ϕX (u1 , . . . , un ). The explicit result, which we state here for completeness, is in fact the n-dimensional counterpart of Proposition 2.20 and is expressed by the relation 1 P(ak < Xk ≤ bk ) = lim c→∞ (2π )n ×

n

k=1

c

c ···

−c

−c

eiuk ak − eiuk bk iuk

ϕX (u1 , . . . , un ) du1 · · · dun (3.30)

where the real numbers ak , bk (k = 1, . . . , n) delimitate a bounded parallelepiped (i.e. an interval in Rn ) whose boundary has zero probability measure. 3.3.1

3.3.1.1

Additional remarks: the multi-dimensional case and the practical calculation of moments The multi-dimensional case

In the preceding section we have been mainly concerned with twodimensional random vectors but it is reasonable to expect that most of the considerations can be readily extended to the n-dimensional case. We only outline this extension here because it will not be difﬁcult for the reader to ﬁll in the missing details. It is implicitly assumed, however, that the reader has some familiarity with matrix notation and basic matrix properties (if

The multivariate case

91

not, one may refer, for example, to Chapter 11 of Ref. [3] at the end of this chapter, to the excellent booklet [16] or to the more advanced text [8]). Given a n-dimensional random vector X = (X1 , . . . , Xn ) and a positive integer k, the kth order moments are deﬁned as m(k1 , k2 , . . . , kn ) = E X1k1 X2k2 . . . Xnkn

(3.31)

where k1 , . . . , kn are n non-negative integers such that k = k1 +k2 +· · ·+kn . This implies that we have now n ﬁrst-order moments – which can be denoted m1 , m2 , . . . , mn – and n2 second-order ordinary and central moments. These latter quantities are often conveniently arranged in the so-called covariance matrix ⎛

K11 ⎜K21 ⎜ K=⎜ . ⎝ ..

K12 K22 .. .

Kn1

Kn2

⎞ . . . K1n . . . K2n ⎟ ⎟ T .. ⎟ = E[(X − m)(X − m) ] .. . ⎠ . . . . Knn

(3.32a)

where Kij = Cov(Xi Xj ) with i, j = 1, . . . , n and in the second expression (X−m) is the n×1 column matrix whose elements are X1 −m1 , . . . , Xn −mn , that is, the difference of the two column matrices X = (X1 , . . . , Xn )T and the ﬁrst-order moments matrix m = (m1 , . . . , mn )T . In this light, it is easy to notice that the covariance matrix can be written as K = E(XXT ) − mmT

(3.32b)

The matrix K is obviously symmetric (i.e. Kij = Kji or, in matrix symbolism, K = KT ) so that there are only n(n + 1)/2 distinct elements; also it is evident that the elements on the main diagonal are the variances of the individual r.v.s – that is, Kii = Var(Xi ). Similar considerations of symmetry and of number of distinct elements apply to the correlation matrix R deﬁned as ⎛

1 ⎜ρ21 ⎜ R=⎜ . ⎝ ..

ρ12 1 .. .

ρn1

ρn2

⎞ . . . ρ1n . . . ρ2n ⎟ ⎟ .. ⎟ .. . ⎠ . ... 1

(3.32c)

where (eq. (3.22)) ρij = Kij /σi σj and, for brevity, we denote by σi = √ Var(Xi ) the standard deviation of the r.v. Xi (assuming that σi is ﬁnite for each i = 1, . . . , n). The relation between K and R – as it is immediately

92 Probability theory veriﬁed by using the rules of matrix multiplication – is K = SRS

(3.33)

where we called S = diag(σ1 , σ2 , . . . , σn ) the matrix whose the only nonzero elements are σ1 , . . . , σn on the main diagonal. If the n r.v.s are pairwise uncorrelated – or, which is a stronger condition, pairwise independent – then both K and R are diagonal matrices and in particular R = I, where I is the identity, or unit, matrix (its only non-zero elements are ones on the main diagonal). Clearly, this holds true if the r.v.s Xi are mutually independent; in this case we can also generalize Proposition 3.4 on ﬁrst-order moments to the multiplication rule n n

Xi = E(Xi ) (3.34) E i=1

i=1

If we pass from the random vector X to a m-dimensional random vector Y by means of a linear transformation Y = AX – where A is a m × n matrix of real numbers – we can use the second expression of (3.32a) to determine Y the covariance matrix KY of Y in terms of KX . In fact, calling for brevity the ‘centered’ matrices Y − mY and X − mX , respectively, we get and X T] Y YT ) = E[(AX)(A X) KY = E( X T AT ] = A E(X X T )AT = A KX AT = E[AX

(3.35)

where we used the well-known relation stating that the transpose of a product of matrices equals the product of the transposed matrices taken in reverse T AT . We mention here in passing a T =X order – that is, in our case (AX) ﬁnal property of the covariance and correlation matrices: both K and R are positive semi-deﬁnite. As it is known from matrix theory, this means that zT Kz ≥ 0

(3.36)

where z is a column vector of n real or complex variables and xT Kx is the so-called quadratic form of the (symmetric) matrix K. Equation (3.36) implies that det(K) – that is, the determinant of K, often also denoted by |K| – is non-negative. Clearly, the same considerations apply to R. The characteristic function of a n-dimensional random vector is the straightforward extension of eq. (3.24b) and the generalization of eq. (3.26) reads k ϕ (u) ∂ X (3.37) ik m (k1 , . . . , kn ) = k1 kn ∂u · · · ∂un 1

u=0

provided that the moment of order k = k1 + k2 + · · · + kn exists.

The multivariate case

93

The marginal CFs can be obtained from ϕX (u1 , . . . , un ) as stated in Section 3.3. For example, ϕX (u1 , u2 , . . . , un−1 , 0) is the joint-CF of the vector (X1 , . . . , Xn−1 ), ϕX (u1 , 0, . . . , 0) is the one-dimensional CF of the r.v. X1 , etc., and the multiplication rule ϕX (u1 , . . . , un ) =

n

ϕXi (ui )

(3.38)

i=1

is a necessary and sufﬁcient condition for the mutual independence of the r.v.s Xi (i = 1, . . . , n). Similarly, all the other considerations apply. In addition, we can determine how a joint-CF changes under a linear transformation from X to a m-dimensional random vector Y. As above, the transformation is expressed in matrix form as Y = AX and we assume here that ϕX (u) is known so that ϕY (v) = E[eiv

T

Y

= E[ei(A

T

] = E[eiv T

v) X

T

AX

]

] = ϕX (AT v)

(3.39a)

In the more general case Y = AX + b – where b = (b1 , . . . , bm )T is a column vector of constants – then it is immediate to determine T

ϕY (v) = ei v b ϕX (AT v)

(3.39b)

In the preceding section nothing has been said about moment-generating functions (MGFs) but by now it should be clear that the deﬁnition is MX (s1 , . . . , sn ) = E[exp(sT X)]

(3.40)

where s1 , . . . , sn is a set of n variables. Within the limitations on the existence of MX outlined in Section 2.4 we have the n-dimensional version of eq. (2.48), that is, ∂ k MX (s) m(k1 , . . . , kn ) = k (3.41) ∂s 1 · · · ∂sknn 1

3.3.1.2

s=0

The practical calculation of expectations

Many quantities introduced so far – moments in the ﬁrst place but CFs and MGFs as well – are deﬁned as expectations, which means, by deﬁnition, as abstract Lebesgue integral in dP. Therefore, the problem arises of how these integrals can be calculated in practice. With the additional slight complication of n-dimensionality, the general line of reasoning parallels closely all that has been said in Chapter 2. We will brieﬂy repeat it here.

94 Probability theory Given a random vector X on the probability space (W, S, P) – that is, a measurable function X : W → Rn – we can make probability statements regarding any Borel set B ∈ Bn by considering the probability measure PX deﬁned by eq. (3.1) and working in the induced real probability space (Rn , Bn , PX ). This is the space of interest in practice, PX being a Lebesgue– Stieltjes measure (on Rn ) which, by virtue of eq. (3.2), can be associated with the PDF FX . At this point we note that the n-dimensional version of Proposition 2.17 applies, with the consequence that our abstract Lebesgue integral on W can be calculated as a Lebesgue–Stieltjes integral on Rn . Then, depending on the type of random vector under study – or, equivalently, on the continuity properties of FX – this integral turns into a form amenable to actual calculations. As in the one-dimensional case, there are three possible cases: the discrete, the absolutely continuous and the singular continuous case, the ﬁrst two (or a mixture thereof) being by far the most important in practice. If X is a discrete random vector its range is a discrete subset AX ⊂ Rn and its complete probabilistic description can be given in terms of the mass distribution pX (x) (see also eq. (3.4)) which, in essence, is a ﬁnite or countable set of real non-negative numbers pi1 ,i2 ,...,in (the n indexes i1 , . . . , in mean that the ith r.v. Xi can take on the values xi1 , xi2 , . . .) such that the normalization condition

pX (x) =

pi1 ,...,in = 1

(3.42)

i1 ,...,in

x∈AX

holds. In this light, given a Borel measurable function g(x) its expectation is given by the sum E[g(x)] =

g(x)pX (x)

(3.43)

x∈AX

Also, the marginal mass distribution of any group of any m(m < n) random variables is obtained by summing the pi1 ,i2 ,...,in over all the n − m remaining variables; so, for example, the marginal mass distribution of the vector (X1 , . . . , Xn−1 ) is given by in pi1 ,i2 ,...,in . This is just a straightforward generalization of eq. (3.10a). If X is absolutely continuous there exists a density function fX (x) such that eqs (3.7a–3.7c) hold. Then, the expectation of a measurable function g(x) becomes a Lebesgue integral and reads E[g(x)] =

g(x)fX (x) dx Rn

(3.44)

The multivariate case

95

Rn .

where dx is the Lebesgue measure on As one might expect – since the Lebesgue integral is, broadly speaking, a generalization of the ordinary Riemann integral of basic calculus – the integral (3.44) coincides with Riemann’s (when this integral exists). In the multi-dimensional case, however, a result of fundamental importance is Fubini’s theorem (Appendix B) which guarantees that a Lebesgue multiple integral can be calculated as an iterated integral under rather mild conditions on the integrand function. This theorem is the key to the practical evaluation of multi-dimensional integrals. Two ﬁnal comments are in order before closing this section. First, we recall eq. (3.10b) and note that their n-dimensional extension is immediate; in fact, the marginal pdfs of any ‘subvector’ of X of m(m < n) components is obtained by integrating fX with respect to the remaining n − m variables. So, for instance, fX1 (x1 ) = Rn−1 fX (x1 , . . . , xn ) dx2 · · · dxn is the pdf of ∞ the r.v. X1 and fY (x1 , . . . , xn−1 ) = −∞ fX (x1 , . . . , xn ) dxn is the (n − 1)dimensional joint-pdf of the random vector Y = (X1 , X2 , . . . , Xn−1 ). The second comment has to do with the CF of an absolutely continuous random vector X. Owing to eq. (3.44), in fact, ϕX (u) and fX (x) turn out to be a Fourier transform pair so that we have fX (x) ei u

ϕX (u) =

T

x

dx

(3.45a)

Rn

with the inversion formula 1 fX (x) = (2π )n

3.3.2

ϕX (u) e−i u

T

x

du

(3.45b)

Rn

Two important examples: the multinomial distribution and the multivariate Gaussian distribution

A multinomial trial with parameters p1 , p2 , . . . pn is a trial with n possible outcomes where the probability of the ith outcome is pi (i = 1, 2, . . . , n) and, clearly, p1 + p2 + · · · + pn = 1. If we perform an experiment consisting of N independent and identical multinomial trials (so that the pi do not change from trial to trial) we may call Xi the number of trials that result in outcome i so that each Xi is a r.v. which can take on any integer value between zero and N. Forming the vector X = (X1 , . . . , Xn ), its joint-pmf is called multinomial (it can be obtained with the aid of eq. (1.16)) and we have p(x1 , . . . , xn ) =

N! px1 , px2 · · · pxnn x1 !x2 ! . . . xn ! 1 2

(3.46a)

96 Probability theory where, since each xi represents the number of times in which i occurs n

xi = N

(3.46b)

i=1

The name multinomial is due to fact that the expression on the r.h.s. of (3.46a) is the general term in the expansion of (p1 + · · · + pn )N . If n = 2 eq. (3.46a) reduces to the binomial pmf considered in Examples 2.8 and 2.9 (a word of caution on notation: in eq. (2.41a) n is the total number of trials while here n is the number of possible outcomes in each trial). As an easy example we can consider three throws of a fair die. In this case N = 3 and n = 6, xi is the number of times the face 1 shows up, x2 is the number of times the face 2 shows up, etc. and p1 = · · · = p6 = 1/6. As a second example we can think of a box with, say, 50 balls of which 10 are white, 22 yellow and 18 are red. The experiment may consist in extracting – with replacement – 5 balls from the box and then counting the extracted balls of each color. In this case N = 5, n = 3 and p1 = 0.20, p2 = 0.44, p3 = 0.36. Note that after each extraction it is important to replace the ball in the box, otherwise the probabilities pi would change from trial to trial and one of the basic assumptions leading to (3.46) would fail. The joint-CF of the multinomial distribution is obtained from eq. (3.24b) by noting that in this discrete case the Lebesgue–Stieltjes integral deﬁning the expectation becomes a sum on all the xi s. Therefore

N! px1 · · · pxnn ei(u1 x1 +···+un xn ) x1 ! . . . xn ! 1 N! (p1 eiu1 )x1 · · · (pn eiun )xn = x1 ! . . . xn !

ϕX (u1 , . . . , un ) =

(3.47)

= (p1 eiu1 + · · · + pn eiun )N As a second step, let us obtain now the marginal CF of one of the r.v.s Xi , for example, X1 . In order to do this (recall Sections 3.2 and 3.3) we must set u2 = u3 = · · · = un = 0 in eq. (3.47) thus obtaining ϕX1 (u1 ) = (p1 eiu1 + p2 + · · · + pn )N = (1 − p1 + p1 ei u1 )N

(3.48)

which is the CF of a one-dimensional binomial r.v. (eq. (2.51)). Also, using the ﬁrst of (2.47b) we can obtain the ﬁrst moment of X1 , that is, E(X1 ) = Np1

(3.49)

in agreement with eq. (2.41b) and with the result we would get by using eq. (3.37a) and calculating the derivative ∂ϕX (u)/∂u1 |u=0 of the joint-CF

The multivariate case

97

(3.47). Clearly, by substituting the appropriate index, both eqs (3.48) and (3.49) apply to each one of the Xi . At this point one may ask about the calculation of E(X1 ) by directly using eq. (3.43) without going through the CF. This calculation is rather cumbersome but we outline it here for the interested reader. We have E(X1 ) =

all xi

N x1 x1 N! x1 px11 · · · pxnn = N(1 − p1 )N−1 p x1 ! . . . xn ! x1 ! 1 x1 =0

= Np1 (1 − p1 )N−1

N x1 =1

(3.50)

x1 x1 −1 p x1 ! 1

where we ﬁrst isolated the sum over x1 , then used the multinomial theorem for the indexes 2, 3, . . . , N and took into account that p2 + p3 + · · · + pn = 1 − p1 . Then, starting from the multinomial theorem all xi

N! px1 · · · pxnn = (p1 + p2 + · · · + pn )N x1 ! · · · xn ! 1

we can differentiate both sides with respect to p1 and then, on the l.h.s. of the resulting relation, isolate the sum on x1 . This procedure leads in the end to (1 − p1 )N−1

N x1 x1 −1 p =1 x1 ! 1

x1 =1

which, in turn, can be substituted in the last expression of (3.50) to give the desired result E(X1 ) = Np1 . After this, it is evident that the shortest way to determine the covariance between any two r.v.s Xk , Xm (where k, m are two integers < n with k = m) is by using the CF. If we recall that Cov(Xk Xm ) = E(Xk Xm ) − E(Xk )E(Xm ) then we only need to calculate the ﬁrst term on the r.h.s. because, owing to (3.49), the second term is N 2 pk pm . Performing the prescribed calculations we get ∂ 2 ϕX (u) E(Xk Xm ) = − = N(N − 1)pk pm (3.51) ∂uk ∂um u=0

and therefore the off-diagonal terms of the covariance matrix K are given by Cov(Xk Xm ) = −Npk pm

(3.52)

By the same token, for any index 1 ≤ k ≤ n, it is not difﬁcult to obtain E(Xk2 ) = Npk [(N − 1)pk + 1] so that the elements on the main diagonal

98 Probability theory of K are Var(Xk ) = Npk (1 − pk )

(3.53)

At ﬁrst sight – since we spoke of independent trials – the fact that the variables Xi are correlated may seem a bit surprising. The correlation is due to the ‘constraint’ eq. (3.46b) and the covariances are negative (eq. (3.52)) because an increase of any one xi tends to decrease the others. The fact that there exists one constraint equation on the xi implies that K is singular (i.e. det(K) = 0) and has rank n − 1 so that, in essence, the n-dimensional vector X belongs to the (n − 1)-dimensional Euclidean space. Let us consider now the multi-dimensional extension of the Gaussian (or normal) probability law considered in Examples 2.4, 2.8 and 2.9(b). For simplicity, we begin with the two-dimensional case. In the light of eq. (3.16b), the joint-pdf of two independent and individually normal r.v.s X, Y forming a vector X must be # " 1 1 (x − m1 )2 (y − m2 )2 fX (x, y) = (3.54) exp − + 2π σ1 σ2 2 σ12 σ22 where m1 = E(X), σ12 = Var(X) and m2 = E(Y), σ22 = Var(Y). Also, using the result of eqs (2.52) and (3.28) of Proposition 3.5, the joint-CF of X is * + 1 2 2 2 2 ϕX (u, v) = exp i(um1 + vm2 ) − (σ1 u + σ2 v ) 2

(3.55)

which is easy to cast in matrix form as

1 ϕX (u) = exp iu m − uT Ku 2

T

(3.56)

where, in the present case, it should be noted that K = diag(σ12 , σ22 ) because of independence – and therefore uncorrelation – between X and Y. The matrix form of the pdf (3.54) is a bit more involved but only a small effort is required to show that we can write fX (x) =

1 exp − (x − m)T K−1 (x − m) 2 2π det(K) ,

1

(3.57)

, where det(K) = σ1 σ2 and K−1 = diag(1/σ12 , 1/σ22 ). From the vector X we can pass to the vector Z of standardized normal r.v.s by means of the linear transformation Z = S−1 (X − m), where S is the diagonal matrix introduced in eq. (3.33). By virtue of eq. (3.39b) the CF

The multivariate case

99

of Z is −1/2 vT R v

ϕZ (v) = e

*

+

1 = exp − (v12 + v22 ) 2

(3.58)

where R is the correlation matrix which, in our case of independent r.v.s, equals the identity matrix I = diag(1, 1). As expected, the CF (3.58) is the product of two standardized one-dimensional CFs and, clearly, the joint-pdf will also be in the form of product of two standardized one-dimensional pdfs, that is, fZ (z1 , z2 ) = fZ1 (z1 )fZ2 (z2 ) =

* + 1 2 1 exp − z1 + z22 2π 2

(3.59)

If, on the other hand, the two normal variables X, Y are correlated eqs (3.56) and (3.57) are still valid but it is understood that now K – and therefore K−1 – are no longer diagonal because K12 = Cov(X, Y) = 0. So, the explicit expression of the joint-CF becomes * + 1 2 2 ϕX (u, v) = exp i(u m1 + v m2 ) − σ1 u + σ22 v2 + 2K12 uv (3.60) 2 from which – by setting u = 0 or v = 0, as appropriate – it is evident that both marginal distributions are one-dimensional CFs of Gaussian random variables (see eq. (2.52)). Using eq. (3.57), the explicit form of the joint-pdf is written fX (x, y) =

1 e−γ (x,y)/2 , 2π σ1 σ2 (1 − ρ 2 )

(3.61a)

where ρ = K12 /σ1 σ2 is the correlation coefﬁcient between X and Y and the function γ (x, y) in the exponential is 1 γ (x, y) = 1 − ρ2

"

(x − m1 )2 σ12

# (x − m1 )(y − m2 ) (y − m2 )2 − 2ρ + σ1 σ2 σ22 (3.61b)

From eqs (3.60) and/or (3.61) we note an important property of jointly Gaussian random variables: the condition Cov(X, Y) = 0 – and therefore ρ = 0 if both σ1 , σ2 are ﬁnite and different from zero – is necessary and sufﬁcient for X and Y to be independent. In this case, in fact, the jointCF (pdf) becomes the product of two one-dimensional Gaussian CFs (pdfs). The equivalence of uncorrelation and independence for Gaussian r.v.s is noteworthy because – we recall from Section 3.3 – uncorrelation does not, in general, imply independence.

100 Probability theory We consider now another important result for jointly Gaussian r.v.s. Preliminarily, we notice that – referring to a three-dimensional system of coordinate axes x, y, z – the graph of the pdf,(3.61) is a bell-shaped surface with maximum of height z = (2π σ1 σ2 1 − ρ 2 )−1 above the point x = m1 , y = m2 . If we cut the surface with a horizontal plane (i.e. parallel to the x, y-plane), we obtain an ellipse whose projection on the x, y-plane has equation (x − m1 )2 σ12

− 2ρ

(x − m1 )(y − m2 ) (y − m2 )2 + = const. σ1 σ 2 σ22

(3.62)

which, in particular, is a circle whenever ρ = 0 and σ1 = σ2 . On the other hand, as ρ approaches +1 or −1, the ellipse becomes thinner and thinner and more and more needle-shaped until ρ = 1 or ρ = −1, when the ellipse degenerates into a straight line. In these limiting cases det(K) = 0, K−1 does not exist and one variable depends linearly on the other. In other words, we are no longer dealing with a two-dimensional random vector but with a single random variable and this is why one speaks of degenerate or singular Gaussian distribution. Returning to our main discussion, the important result is the following: when the principal axes of the ellipse are parallel to the coordinate axes, then ρ = 0 and the two r.v.s are uncorrelated – and therefore independent. In other words, by means of a rotation of the coordinate axes x, y it is always possible to pass from a pair of dependent Gaussian variables – whose pdf is in the form (3.61) – to a pair of independent Gaussian variables. This property can be extended to n dimensions and is frequently used in statistical applications (see the following chapters). Let us examine this property more closely. Given the ellipse (3.62), it is known from analytic geometry that the relation tan 2α =

2ρσ1 σ2 σ12 − σ22

(3.63)

determines the angles α between the x-axis and the principal axes of the ellipse (eq. (3.63) leads to two values α, namely α1 , α2 where α1 , α2 differ by π/2). If we rotate the x, y-plane through an angle α, the new coordinate axes are parallel to the ellipse principal axes and the cross-product term in (3.62) vanishes. If, in addition to this rotation, we perform now a rigid translation of the coordinate system which brings the origin to the point (m1 , m2 ), the ellipse will also be centered in the origin. At this point, the original pdf (3.61) has transformed into 1 p2 q2 f (p, q) = (3.64a) exp − 2 − 2π σp σq 2σq2 2σp

The multivariate case

101

where we call p, q the ﬁnal coordinate axes obtained by ﬁrst rotating and then rigidly translating the original axes x, y. Moreover, it can be shown that the new variances σp2 , σq2 are expressed in terms of the original variances by the relations σp2 = σ12 cos2 α + ρσ1 σ2 sin 2α + σ22 sin2 α σq2 = σ12 sin2 α − ρσ1 σ2 sin 2α + σ22 cos2 α

(3.64b)

, from which it follows σp σq = σ1 σ2 1 − ρ 2 . Equation (3.64b) is obtained by noticing that the (linear, since α is ﬁxed) relation between the x, y and the p, q axes – and therefore between the original random vector X = (X, Y)T and the new random vector P = (P, Q)T – is p cos α = q − sin α

sin α cos α

x m1 − m2 y

(3.65)

so that eq. (3.64b) follows by virtue of eq. (3.35). As a side comment, eqs (3.64b) represent the diagonal terms of the covariance matrix of the twodimensional vector P. By using eqs (3.35) and (3.63) the reader is invited to determine that, as expected, Cov(P, Q) = 0, that is, that the off-diagonal terms of the ‘new’ covariance matrix are zero. From (3.64a), if needed, it is then possible to take a further step and pass to the standardized Gaussian random vector Z whose pdf and CF are given by eqs (3.59) and (3.58), respectively. At this point we can consider a frequently encountered problem and determine the probability Pk that a point falls within the ellipse whose principal axes are k times the standard deviations σp , σq of the two variables. Calling Ek this ellipse (centered in the origin), the probability we are looking for is Pk = P[(P, Q) ∈ Ek ] =

f (p, q) dp dq

(3.66a)

Ek

where f (p, q) is given by eq. (3.64a). Passing to the standardized variables z1 = p/σp and z2 = q/σq the ellipse Ek becomes a circle Ck of radius k and 2 2 % $ z z 1 dz1 dz2 exp − 1 − 2 Pk = P Z12 + Z22 ≤ k2 = 2π 2 2

(3.66b)

Ck

The integral on the r.h.s. can now be calculated by turning to the polar coordinates z1 = r cos θ, z2 = r sin θ (recall from analysis that the Jacobian

102 Probability theory determinant of this transformation is |J| = r) and we ﬁnally get 1 Pk = 2π

2π k

−(r2 /2)

re 0 0

k dr dθ =

r e−(r

2

/2)

dr = 1 − exp(−k2 /2)

0

(3.66c) so that, for instance, we have the probabilities P1 = 0.393, P2 = 0.865 and P3 = 0.989 for k = 1, k = 2 and k = 3, respectively. This result is the twodimensional counterpart of the well-known fact that for a one-dimensional Gaussian variable X the probability of obtaining a value within k standard deviations is ⎧ ⎪ ⎨0.683 for k = 1 (3.67) P[|X − µX | ≤ kσX ] = 0.954 for k = 2 ⎪ ⎩ 0.997 for k = 3 Equation (3.67), in addition, can also be used to calculate the twodimensional probability to obtain a value of the vector (P, Q) within the rectangle Rk of sides 2kσp , 2kσq centered in the origin. In fact, since P and Q are independent, the two-dimensional probability P[(P, Q) ∈ Rk ] is given by the product of the one-dimensional probabilities (3.67); consequently P[(P, Q) ∈ R1 ] = (0.683)2 = 0.466, P[(P, Q) ∈ R2 ] = (0.954)2 = 0.910, etc. and it should be expected that P[(P, Q) ∈ Rk ] > P[(P, Q) ∈ Ek ] because the ellipse Ek is inscribed in the rectangle Rk . We close this rather lengthy section with a few general comments on Gaussian random vectors in any number of dimensions: (i) The property of being Gaussian is conserved under linear transformations (as the discussion above has shown more than once in the two-dimensional case). (ii) The marginal distributions of a jointly-Gaussian are individually Gaussian. However, the reverse may not be true and examples can be given of individually Gaussian r.v.s which, taken together, do not form a Gaussian vector. (iii) The CF and pdf of a jointly-Gaussian n-dimensional vector are written in matrix form as in eqs (3.56) and (3.57); in this latter equation, however, the factor 2π at the denominator becomes (2π )n/2 . √ In other words, at the denominator of (3.57) there must be a factor 2π for each dimension. (iv) Let us examine in the general case the possibility of passing from a Gaussian vector of correlated random variables (i.e. with a nondiagonal covariance matrix) to a Gaussian vector of independent – or even standardized – random variables (that is with a diagonal covariance

The multivariate case

103

matrix). In two-dimensional this was accomplished by ﬁrst rotating the coordinate axes and then translating the origin to the point (m1 , m2 ) but it is evident that we can ﬁrst translate the axes and then rotate them without changing the ﬁnal result. So, starting from a n-dimensional Gaussian vector X of correlated r.v.s X1 , . . . , Xn whose pdf is

1 1 T −1 fX (x) = (3.68a) exp − (x − m) K (x − m) , 2 (2π )n/2 det(K) -= we can translate the axes and consider the new ‘centered’ vector X X − m with pdf

1 1 T −1 ˆ = fX (3.68b) exp − xˆ K xˆ , - (x) 2 (2π )n/2 det(K) Now, since K is symmetric and positive deﬁnite (i.e. the Gaussian vector is assumed to be non-degenerate), a theorem of matrix algebra states that there exists a non-singular matrix H such that HHT = K. From this relation we get HHT K−1 = I and then HHT K−1 H = H which, in turn, implies HT K−1 H = I. Using this same matrix H let us now pass to the new random - = HZ. The term at the vector Z = (Z1 , . . . , Zn )T deﬁned by the relation X exponential of (3.68b) becomes xˆ T K−1 xˆ = (Hz)T K−1 Hz = zT HT K−1 Hz = zT Iz = zT z which is the sum of squares z12 + z22 + · · · + zn2 . Moreover, the Jacobian determinant of the transformation to Z is det(H) , so that the multiplying factor before the exponential becomes det(H)/ (2π )n det(K); however, from HT K−1 H = I we get,(det H)2 det(K−1 ) = 1 and since det(K−1 ) = [det(K)]−1 , then det(H) = det(K). Consequently, our ﬁnal result is

1 1 T z exp − z (3.69) fZ (z) = 2 (2π )n/2 which, as expected, is the pdf of a standardized Gaussian vector whose covariance matrix is I. Also, it is now clear that the matrix H represents a n-dimensional rotation of the coordinate axes.

3.4

More on conditioned random variables

The subject of conditioning has been discussed in both Chapters 1 and 2 (see, in particular, Section 2.5.2). Here we return on the subject for two main reasons: ﬁrst, because some more remarks are worthy of mention in their own right and, second, because a number of new aspects are due the developments of the preceding sections of this chapter.

104 Probability theory Let us start with some additional results to what has been said in Section 2.5.2 by working mostly in the real probability space (R, B, PX ). We do so because, as stated before, it is (R, B, PX ) which is considered in practice while the original space (W, S, P) is an entity in the background with only occasional interest in applications. Consider an absolutely continuous r.v. X deﬁned on (W, S, P). This, we recall, implies that the measure PX is absolutely continuous with respect to the Lebesgue measure on the real line and there exists a function fX (the pdf of X) such that PX (B) = P(X

−1

(B)) =

fX (x) dx B

Then, although the set C = {x} is indeed a Borel set of R, we cannot – at least in the usual way – deﬁne a conditional probability with respect to this event because PX (x) = 0. However, for h > 0 consider the Borel set Bh deﬁned as Bh = (x − h, x + h]. Then PX (Bh ) > 0 and we can condition on this event by deﬁning, for every A ∈ S, the measure PBh exactly as we did in Section 2.5.2 (eq. (2.60); again with a slight misuse of notation because Bh is not a set of S. Rigorously, we should write PX−1 (Bh ) ). Now, with a slight change of notation let us call PX (· | Bh ) its image measure in R instead of PX | Bh . Then, given B ∈ B we have PX (B | Bh ) =

PX (B ∩ Bh ) PX (Bh )

(3.70)

and at this point we can try to deﬁne PX (B | x) as the limit of PX (B | Bh ) as h → 0. By virtue of Bayes’ theorem (eq. (1.14)) together with the total probability formula of eq. (1.12) we can write PX (B | Bh ) as PX (B | Bh ) = PX (B) = PX (B)

PX (Bh | B) PX (Bh ) FX | B (x + h) − FX | B (x − h) FX (x + h) − FX (x − h)

(the second equality is due to the basic properties of the PDFs FX | B and FB ), then, dividing both the numerator and denominator by 2h and passing to the limit we get the desired result PX (B | x) = PX (B)

fX | B (x) fX (x)

(3.71)

provided that the conditional density fX | B exists. The fact that PX (B | x) is not deﬁned whenever fX (x) = 0 is not a serious limitation because the set

The multivariate case

105

N = {x : fX (x) = 0} has probability zero, meaning that, in practice, it is unimportant as far as probability statements are concerned. In fact PX (N) =

fX (x) dx = 0 N

If now in eq. (3.71) we move the factor fX (x) to the left-hand side and integrate both sides over the real line we get (since fX | B is normalized to unity) ∞ PX (B) =

fX (x) PX (B | x) dx

(3.72)

−∞

which, because of its analogy with eq. (1.12), is the continuous version of the total probability formula (see also eq. (3.80b)); the sum becomes now an integral and the probabilities of the conditioning events Aj are now the inﬁnitesimal probabilities fX (x) dx. In the case of discrete r.v.s the complications above do not exist. If AX is the (discrete) set of values taken on by the r.v. X and xi ∈ AX is such that PX {xi } = pX (xi ) > 0, then the counterpart of eq. (3.71) is PX (B | xi ) = PX (B)

pX | B (xi ) pX (xi )

(3.73)

and it is not deﬁned if p(xi ) = 0. On the other hand, the total probability formula reads PX (B) = pX (xi )PX (B | xi ) (3.74) xi ∈AX

We turn now to some new aspects of conditional probability brought about by the discussion of the previous sections. Consider a two-dimensional absolutely continuous random vector X = (X, Y) with joint-PDF FXY (x, y) and joint-pdf fXY (x, y); we want to determine the statistical description of, say, Y conditioned on a value taken on by the other variable, say X = x. We have now three image measures, PX , PY in R and the joint measure PXY (or PX ) in R2 ; all of them, however, originate from P in W. Therefore, if we look for a probabilistic description of an event relative to Y conditioned on an event relative to X it is reasonable to consider – in (W, S) – the conditional probability (where Jy = (−∞, y] and Bh is as above) P[Y −1 ( Jy ) | X −1 (Bh )] =

P[Y −1 ( Jy ) ∩ X −1 (Bh )] PXY ( Jy ∩ Bh ) = PX (Bh ) P(X −1 (Bh ))

106 Probability theory and deﬁne the conditional PDF FY | X (y | x) as the limit of this probability as h → 0. As before, we divide both the numerator and denominator by 2h and pass to the limit to get FY | X (y | x) =

1 fX (x)

∂FXY (x, y) ∂x

(3.75a)

Then, taking the derivative of both sides with respect to y we obtain the conditional pdf fY | X (y | x) =

1 fX (x)

fXY (x, y) ∂ 2 FXY (x, y) = fX (x) ∂y∂x

(3.75b)

A few remarks are in order: (a) the function fY | X is not deﬁned at the points x where fX (x) = 0; however, as noticed above, this is not a serious limitation; (b) fY | X is a function of y alone and not a function of the two variables x, y. In this case x plays the role of a parameter: for a given value, say x1 , we have a function fY | X (y | x1 ) and we have a different function fY | X (y | x2 ) for x2 = x1 ; (c) being a pdf in its own right, fY | X is normalized to unity. In fact, recalling eq. (3.10b) we get ∞ −∞

1 fY | X (y | x) dy = fX (x)

∞ fX (x, y) dy = 1 −∞

For the same reason it is clear that the usual relation between pdf and PDF holds, that is y FY | X (y | x) =

fY | X (t | x) dt −∞

(d) the symmetry between the two variables leads immediately to the conditional-pdf fX | Y (x | y) of X given Y = y, that is, fX | Y (x | y) =

fX (x, y) fY (y)

(3.76)

(e) if X is discrete and AX = {x1 , x2 , . . .}, AY = {y1 , y2 , . . .} are the ranges of X and Y, respectively, then the joint-pmf takes on values in AX × AY

The multivariate case

107

and the counterpart of (3.76) can be written as

pX | Y (xi | yk ) = P(X = xi | Y = yk ) =

pX (xi , yk ) pY (yk )

(3.77)

where it is assumed that pY (yk ) = 0.

In the light of the considerations above, we can now obtain some relations which are often useful in practical cases. We do so for the absolutely continuous case leaving the discrete case to the reader. First, by virtue of eqs (3.75b) and (3.76) we note that it is possible to express the joint-pdf of X in the two forms fXY (x, y) = fY | X (y | x)fX (x) fXY (x, y) = fX | Y (x | y)fY (y)

(3.78)

Then, combining these two results we get

fX | Y (x | y) = fY | X (y | x)

fX (x) fY (y)

(3.79)

and a similar equation for fY | X . Next, in order to obtain the counterpart of the total probability expression of eq. (3.72), we can go back to the probability P by letting B be the event Y −1 ( Jy ); then P(Y −1 ( Jy )) = FY (y) and we obtain the marginal-PDF of Y in terms of the conditional-PDF FY | X and of the pdf of the conditioning variable, that is, ∞ FY (y) =

FY | X (y | x)fX (x) dx

(3.80a)

−∞

Differentiating with respect to y on both sides leads to the total probability formula ∞ fY (y) =

fY | X (y | x)fX (x) dx −∞

(3.80b)

108 Probability theory (which could also be obtained from the second of (3.10b) using (3.79)). By symmetry, it is then evident that ∞ FX | Y (x | y)fY (y) dy

FX (x) = −∞

(3.81)

∞ fX (x) =

fX | Y (x | y)fY (y) dy −∞

If the expression (3.80b) for fY is inserted at the denominator of eq. (3.79) we note a formal analogy with Bayes’ theorem of eq. (1.14); for this reason eq. (3.79) – and its counterpart for fY | X – is also called Bayes’ formula (in the continuous case). In Section 3.2.1 we pointed out that two variables X, Y are independent if and only if fXY (x, y) = fX (x)fY (y). Therefore, by virtue of eq. (3.78), independence implies fY | X (y | x) = fY (y)

(3.82)

fX | Y (x | y) = fX (x)

as it might be expected considering that knowledge of a speciﬁc outcome, say X = x, gives no information on Y. At this point, the extension to more than two r.v.s is immediate and we a multionly mention it brieﬂy here, leaving the rest to the reader. If we call X dimensional vector of components X1 , . . . , Xm , Y1 , . . . , Yn with joint-pdf fX then, for example, fX | Y (x1 , . . . , xm | y1 , . . . , yn ) =

fX (x1 , . . . , xm , y1 , . . . , yn ) fY (y1 , . . . , yn )

(3.83)

where we denoted by fY the marginal-pdf relative to the n Y-type variables. Similarly, the generalization of eq. (3.80b) becomes fY (y) =

fY | X (y | x)fX (x) dx

(3.84)

Rm

where fX the marginal-pdf of the X variables and dx = dx1 · · · dxm . As an exercise to close this section, we also invite the reader to examine the case of a two-dimensional vector X = (X, Y) where X is absolutely continuous with pdf fX (x) and Y is discrete with pmf deﬁned by the values pY (yi ).

The multivariate case 3.4.1

109

Conditional expectation

As noted in Section 2.5.2, the theory deﬁnes conditional expectations as abstract Lebesgue integrals in W with respect to an appropriate conditional measure which, in turn, depends on the conditioning event and is ultimately expressed in terms of the original measure P. In practice, however, owing to the relation between measures in W and their image measures (through a random variable or a random vector), expectations become in the end Lebesgue–Stieltjes integrals on R, R2 or Rn , whichever is the case. These integrals, in turn, are sums or ordinary Lebesgue integrals (i.e. Riemann integrals in most applications) depending on the type of distribution function. Owing to the developments of the preceding section, it should be expected that the conditional expectation of X given the event Y = y is expressed as E(X | y) =

x dFX | Y

(3.85)

R

which, in the absolutely continuous case becomes ∞ E(X | y) =

xfX | Y (x | y) dx = −∞

1 fY (y)

∞ xfX (x, y) dx

(3.86)

−∞

where in the second equality we took eq. (3.76) into account. It is understood that analogous relations hold for E(Y | x). On the other hand, in the discrete case (conditioning on the event Y = yk ) we have E(X | yk ) =

xi pX | Y (xi | yk ) =

1 xi pX (xi , yk ) pY (yk )

(3.87)

i

all i

More generally, if g is a measurable function of both X and Y we have the fundamental relations (their discrete counterparts are left to the reader) ∞ E[g(X, Y) | y] =

g(x, y)fX | Y (x | y) dx −∞

(3.88)

∞ E[g(X, Y) | x] =

g(x, y)fY | X (y | x) dy −∞

It is evident that eq. (3.86) coincides with the ﬁrst of (3.88) when g(x, y) = x and also that the expressions for all conditional moments can be obtained as special cases of eq. (3.88), depending on which one of the two variables is the conditioning one.

110 Probability theory Being based on the properties of the integral, conditional expectations satisfy all the properties of expectation given in Chapter 2. In particular we mention, for example (i) the conditional expectation of a constant is the constant itself; (ii) if a, b are two constants and X, Y1 , Y2 are random variables then linearity holds, that is, E(aY1 + bY2 | x) = aE(Y1 | x) + bE(Y2 | x); (iii) if Y1 ≤ Y2 then E(Y1 | x) ≤ E(Y2 | x). Now, so far we have spoken of conditional expectations of, say, X given Y = y by tacitly assuming that y is a given, well-speciﬁed value. If we adopt a more general point of view we can look at expectations as functions of the values taken on by the random variable Y. In other words since, in general, we have a value of E(X | y) for every given y we may introduce the real-valued function g(Y) ≡ E(X | Y) deﬁned on the range of Y. This function – which, clearly, takes on the value E(X | y) when Y = y – can be shown to be measurable and therefore it is a random variable itself. In this light it is legitimate to ask about its expectation E[g(Y)] = E[E(X | Y)]. The interesting result is that we get E[E(X | Y)] = E(X)

(3.89a)

and, by symmetric arguments E[E(Y | X)] = E(Y)

(3.89b)

In fact, in the absolutely continuous case, for example,

E[E(X | Y)] =

E(X | Y)fY (y) dy =

=

xfX (x, y) dx dy =

xfX | Y (x | y) dx fY (y) dy

x

fX (x, y) dy

dx

=

xfX (x) dx = E(X)

(all integrals are from −∞ to +∞ and eqs (3.76) and (3.10b) have been taken into account). Equations (3.89) – which are sometimes useful in practice – may appear confusing at ﬁrst sight but they state a reasonable fact: for instance, eq. (3.89a) shows that E(X) can be calculated by taking a weighted average on all the expected values of X given Y = y, each term being weighted by the probability of that particular conditioning event Y = y.

The multivariate case

111

Equation (3.89a) can be generalized to E[g(X)] = E[E(g(X) | Y)]

(3.90)

where g(X) is a (measurable) function of X. With the appropriate modiﬁcations, the same obviously applies to (3.89b). By similar arguments, the reader is invited to prove that Var(X) = E[Var(X | Y)] + Var[E(X | Y)]

(3.91)

Var(Y) = E[Var(Y | X)] + Var[E(Y | X)]

(Hint: to prove the ﬁrst of (3.91) start from Var(X | Y) = E(X 2 | Y) − E2 (X | Y) and use eq. (3.90). For our purposes, the discussion above sufﬁces. However, for the interested reader we close this section with some additional remarks of theoretical nature on the function E(X | Y). We simply outline the general ideas and more details can be found in the references at the end of the chapter. Consider an event G ∈ S. As a consequence of eq. (2.58), the expectation of a r.v. X conditioned on G can be written as 1 1 E(X | G) = X dPG = IG X dP = XdP (3.92a) P(G) P(G) W

W

G

which leads to P(G)E(X | G) =

X dP

(3.92b)

G

This last expression makes no reference to the conditional measure PG and can be assumed to be the deﬁning relation of E(X | G). Clearly, in the same = {∅, G, GC , W} is a way one can deﬁne E(X | GC ). Then, noting that G ⊂ S (the σ -algebra generated by G) one can deﬁne a function σ -algebra G on G as E(X | G) = E(X | G)IG + E(X | GC )IGC E(X | G)

(3.93)

is a simple function (see the deﬁnition of simple function in E(X | G) Appendix B) which is measurable – and therefore a random variable – with and it is such that, for every set A ∈ G respect to both S and G A

dP = E(X | G)

X dP A

(3.94)

112 Probability theory (in this case A can only be one of the four sets ∅, G, GC , W; using the deﬁnition of integral for simple functions, the reader is invited to verify eq. (3.94)). If, in particular X = IF (where F ∈ S) then the r.h.s. of (3.94) equals P(F ∩A). then by virtue of the Radon– Setting PF (A) = P(F ∩ A) for every A ∈ G Nikodym theorem we can deﬁne the conditional probability as a special case of conditional expectation, that is, = E(IF | G) P(F | G)

(3.95)

which agrees with the fact that the measure of a set is the expectation of its indicator function. Now, besides this illustrative example, it can be shown that this same line of reasoning extends to any σ -algebra S ⊂ S and the resulting function E(X | S) is called the conditional expectation of X given S. In particular, if S is the σ -algebra generated by a collection of sets G1 , G2 , . . . , Gn ∈ S such that W = ∪ni=1 Gi , then E(X | S) is a r.v. on (W, S, P) which takes on the S. Also, one can value E(X | Gi ) on Gi and satisﬁes eq. (3.94) for every A ∈ deﬁne P(F | S) as above. However, it is not necessary for S to be determined by a ﬁnite collection of sets. Therefore, if Y is another r.v. deﬁned on the space (W, S, P), for every Borel set B ⊂ R one can consider the σ -algebra Y generated by the inverse images Y −1 (B) and introduce the function E(X | Y) which satisﬁes the counterpart of (3.94), that is,

dP = E(X | Y)

Y −1 (B)

X dP

(3.96)

Y −1 (B)

is constant on every set of the form Since it can be shown that E(X | Y) is a Y −1 (y) (where y is a ﬁxed value in R), then it follows that E(X | Y) function of Y which takes on the value E(X | y) for all the elements w ∈ W = E(X) such that w ∈ Y −1 (y). Equation (3.96) then shows that E[E(X | Y)] which, on more theoretical grounds, justiﬁes eq. (3.89a). 3.4.2

Some examples and further remarks

In order to illustrate with an example the considerations of the preceding two sections we start with the bivariate Gaussian distribution. If the two variables X,Y are correlated their joint-pdf is given by eqs (3.61a) and (3.61b). We could obtain the marginal-pdfs by using eq. (3.10b) but it is quicker to consider the joint-CF of eq. (3.60) and note that the marginal CFs are both one-dimensional Gaussian. It follows that fX (x) and fY (y) are Gaussian pdfs with parameters E(X) = m1 , Var(X) = σ12 and E(Y) = m2 , Var(Y) = σ22 , respectively. For the conditional pdfs we can use eq. (3.78) so that, say, fY | X (y | x) is given by fY | X (y | x) = fXY (x, y)/fX (x). Explicitly, after some

The multivariate case

113

manipulations we get fY | X (y | x) =

,

1

σ2 2π(1 − ρ 2 )

exp[−h(x, y)]

(3.97a)

where the function in the exponential is 1 h(x, y) = 2(1 − ρ 2 )

x − m1 y − m2 −ρ σ2 σ1

2 (3.97b)

and can be rewritten in the form h(x, y) =

1 2σ22 (1 − ρ 2 )

2

y − m2 − ρ

σ2 (x − m1 ) σ1

(3.97c)

from which it is evident that the conditional expectation and variance are mY | X = E(Y | X) = m2 + ρ

σ2 (x − m1 ) σ1

(3.98)

σY2 | X = Var(Y | X) = σ22 (1 − ρ 2 ) Equation (3.98) show that (i) as a function of x, the conditional expectation of Y given x is a straight line (which is called the regression line of Y on X) and (ii) the conditional variance does not depend on x. With the obvious modiﬁcations, relations similar to (3.97) and (3.98) hold for fX | Y (x | y), E(X | Y) and Var(X | Y). If the two variables are independent – which, we recall, is equivalent to uncorrelated for the Gaussian case – then the conditional-pdfs coincide with the marginal pdfs and the conditional parameters coincide with the unconditioned ones. Equations (3.97) and their counterparts for fX | Y (x | y), in addition, show that the conditional-pdfs of jointly Gaussian r.v.s are Gaussian themselves. We do not prove it here but it can be shown that this is an important property which extends to the n-dimensional case: all the conditional pdfs that can be obtained from a jointly Gaussian vector are Gaussian. If now, as another example, we consider the joint-pdf of Example 3.1(b) (eq. (3.13a)), the reader is invited to determine that

1 1 fX | Y (x | y) = √ exp − (x + y/2)2 3 3π

(3.99)

E(X | Y) = −y/2 and also that E(XY) = −1. So, if we note from the marginal-pdfs (3.14a) and (3.14b) that E(X) = E(Y) = 0 and Var(X) = Var(Y) = 2, it follows from eqs (3.19a) and (3.22) that Cov(X, Y) = −1 and ρXY = −1/2.

114 Probability theory In the bivariate Gaussian case above we spoke of regression line of Y on X. In the general case of a non-Gaussian pdf E(Y | X) – as a function of x – may not be a straight line and then one speaks of regression curve of Y on X. For some non-Gaussian pdfs, however, it may turn out that E(Y | X) is a straight line, that is, that we have E(Y | X) =

y fY | X (y | x) dy = a + bx

(3.100)

where a and b are two constants. Now, since E[E(Y | X)] = E(Y) we can take eq. (3.100) into account to get E(Y) =

y fXY (x, y) dx dy =

fX (x)

y fY | X (y | x) dy

dx

fX (x)(a + bx) dx = a + bE(X)

=

(3.101)

showing that the regression line passes through the point (E(X), E(Y)). By similar arguments, we also obtain (the easy calculations are left to the reader) E(XY) = aE(X) + bE(X 2 )

(3.102)

so that, in the end, the slope and intercept of the straight line are given by b=

E(XY) − E(X)E(X) Cov(X, Y) = 2 2 Var(X) E(X ) − E (X)

(3.103a)

a = E(Y) − bE(X) where eqs (3.19a) and (3.19b) have been taken into account in the second equality for b. By substituting eqs (3.103a) in (3.100) and recalling eq. (3.22) we see that in all cases where E(Y | X) is a linear function of x the ﬁrst of eq. (3.98) holds. On the other hand, now the second of (3.98) may no longer hold and σY2 | X

=

{y − E(Y | X)}2 f (y | x) dy

is, in general, a function of x. Nonetheless, the quantity σ22 (1 − ρ 2 ) still has a meaning: it represents a measure of the average variability of Y around the regression line on X. In fact, it is left to the reader to show that by deﬁning 2 σY(avg) as the weighted (with the probability density of the x-values) average

The multivariate case of

σY2 | X ,

115

then

2 σY(avg)

≡

σY2 | X fX (x) dx = σ22 (1 − ρ 2 )

(Hint: use (3.100) and the second of (3.103a) to determine that [y − E(Y | X)]2 = (y − E(Y))2 + b2 (x − E(X))2 − 2b(y − E(Y))(x − E(X)), insert in the expression of σY2 | X and then use eq. (3.75b) and the ﬁrst of (3.103a).) 2 From the expression of σY(avg) we note, however, that the second of eq. (3.98) holds whenever σY2 | X does not depend on x. In all these particular cases we 2 have σY(avg) = σY2 | X = σ22 (1 − ρ 2 ). The bivariate Gaussian – as we have seen – is one of these cases. If now, in addition to (3.100), we also assume that E(X | Y) = c + dy then d=

Cov(X, Y) Var(Y)

(3.103b)

a = E(X) − dE(Y) and the geometric mean of b and d is the correlation coefﬁcient, that is, √

bd =

Cov(X, Y) = ρXY σX σ Y

(3.104)

which, as noted in Section 3.3, is a measure of the extent of the linear relationship between the two variables. As a ﬁnal remark we point out that the fact that E(Y | X) is a linear function of x does not necessarily imply, in general, that E(X | Y) is a linear function of y – and conversely; the bivariate Gaussian distribution is, in this respect, an exception. More on linear regression in statistical applications is delayed to Chapter 7.

3.5

Functions of random vectors

It often happens that we have some information on one or more random variables but our interest – rather than in the variables themselves – lies in a function of these variables. In Section 2.5.3, we already touched this subject by considering mainly the one-dimensional case; we now move on from there extending the discussion to random vectors. Let X = (X1 , . . . , Xn ) be a n-dimensional random vector and let Z = (Z1 , . . . , Zn ) be such that Z = g(X), where this symbol means that the

116 Probability theory function g : Rn → Rn has components g1 , . . . , gn and Z1 = g1 (X1 , . . . , Xn ) Z2 = g2 (X1 , . . . , Xn ) .. .

(3.105)

Zn = gn (X1 , . . . , Xn ) First of all we note that Z is a random vector if all the gk (k = 1, . . . , n) are Borel functions, a condition which is generally true in most practical cases. If we suppose further that we are dealing with absolutely continuous vectors and that the joint-pdf fX (x) is known, we can obtain fZ (z) by using a well-known change-of-variables theorem of analysis. This result leads to an equation formally similar to (2.71), that is, fZ (z) = fX (g−1 (z))| det(J)|

(3.106a)

where J is the Jacobian matrix ⎡ −1 ∂g1 /∂z1 ⎢∂g −1 /∂z1 ⎢ 2 J=⎢ .. ⎣ . ∂gn−1 /∂z1

∂g1−1 /∂z2 ∂g2−1 /∂z2 .. . ∂gn−1 /∂z2

⎤ . . . ∂g1−1 /∂zn . . . ∂g2−1 /∂zn ⎥ ⎥ ⎥ .. ... ⎦ . −1 . . . ∂gn /∂zn

(3.106b)

The assumptions of the theorem require that (a) g is one-to-one (so that X1 = g1−1 (Z1 , . . . , Zn ); X2 = g2−1 (Z1 , . . . , Zn ), etc.); (b) all the derivatives are continuous; (c) det( J) = 0. If one (or more) of the gk (k = 1, . . . , n) is not invertible (i.e. not one-toone), one needs to divide the domains of X and Z in a sufﬁcient number – say p – of mutually disjoint subdomains in such a way that – in these subdomains – there exists a one-to-one mapping between the two variables. Then eq. (3.106a) holds in each subdomain and the ﬁnal result fZ (z) is obtained by summing the p contributions. All the results above can also be used if Z is m-dimensional, with m < n. In this case one introduces n–m auxiliary variables and proceeds as stated by the theorem. Provided that the requirements of the theorem are satisﬁed, the choice of the auxiliary variables is arbitrary; therefore it is understood that one should choose them in a way that keeps the calculations as simple as possible. So, for instance, if X = (X1 , X2 ) is a two-dimensional vector and Z = g(X1 , X2 ) is one-dimensional, we can introduce the auxiliary variable

The multivariate case

117

Z2 = X2 ; then, the transformation (3.105) is Z1 = g(X1 , X2 )

(3.107a)

Z2 = X2 and * det( J) = det

∂g −1 /∂z1 0

+ ∂g −1 ∂g −1 /∂z2 = 1 ∂z1

(3.107b)

Consequently, eq. (3.106) reads fZ (z1 , z2 ) = fX (g

−1

∂g −1 (z1 ), x2 ) ∂z1

(3.107c)

and the desired result – that is, the marginal pdf of Z1 – is given by ∞ fZ1 (z1 ) =

fZ (z1 , z2 ) dz2

(3.107d)

−∞

Example 3.3 Let Z = X1 + X2 . As above, we introduce the auxiliary variable Z2 = X2 . Then X1 = Z1 − Z2 X2 = Z2 and det( J) = 1. Therefore fZ (z1 , z2 ) = fX (z1 − z2 , z2 ) and ∞ fZ1 (z1 ) =

∞ fX (z1 − z2 , z2 ) dz2 =

−∞

fX (z1 − x2 , x2 ) dx2

(3.108)

−∞

If the two variables X1 , X2 are independent with pdfs f1 (x1 ), f2 (x2 ), respectively, then fX (x1 , x2 ) = f1 (x1 )f2 (x2 ) and (3.108) becomes ∞ fZ1 (z1 ) =

f1 (z1 − x2 )f2 (x2 ) dx2

(3.109)

−∞

which is called the convolution integral of f1 and f2 ; this is a frequently encountered type of integral in applications of Physics and Engineering and is often denoted by the symbol f1 ∗ f2 . (Incidentally, we note that eq. (3.109) is in agreement with eq. (3.29) on CFs; in fact the Fourier transform of a

118 Probability theory convolution integral is given by the product of the individual Fourier transforms of the functions appearing in the convolution.) So, for instance, if X1 , X2 are independent and are both uniformly distributed in such a way that ' 1/(b − a), a < x ≤ b f1 (x) = f2 (x) = 0, otherwise b eq. (3.109) gives fZ1 (z1 ) = (b − a)−1 a f1 (z1 − x2 ) dx2 where z1 ranges from a minimum value of 2a to a maximum of 2b and is zero otherwise. The integral can be divided into two parts considering that (i) if z1 − x2 > a then x2 < z1 − a and (ii) if z1 − x2 < b then x2 > z1 − b. In the ﬁrst case we have 1 fZ1 (z1 ) = (b − a)2

z1 −a

dx2 = a

z1 − 2a (b − a)2

(3.110a)

which holds for 2a < z1 ≤ a + b (the second inequality is due to the fact that we must have z1 − a ≤ b, therefore z1 ≤ a + b). In the second case 1 fZ1 (z1 ) = (b − a)2

b dx2 = z1 −b

2b − z1 (b − a)2

(3.110b)

which holds for a + b < z1 ≤ 2b (z1 − b > a implies z1 > a + b). The distribution given by eqs (3.110a) and (3.110b) is called Simpson’s distribution. If, turning to another case, X1 , X2 are jointly-Gaussian and not independent (eqs (3.61a) and (3.61b)) then it can be shown (Refs [3, 4, 6, 17]) that the pdf of the r.v. Z = X1 + X2 is 1 (z − m1 − m2 )2 fZ (z) = ) exp − 2(σ12 + 2ρσ1 σ2 + σ22 ) 2π(σ12 + 2ρσ1 σ2 + σ22 ) (3.111) which is also Gaussian. The reverse statement, in general, is not true and the fact that Z = X1 + X2 is Gaussian does not necessarily imply that X1 , X2 are individually Gaussian. It does, however, if X1 , X2 are independent (Cramer’s theorem). In this case, fZ (z) is obtained by simply setting ρ = 0 in eq. (3.111). All these considerations on jointly-Gaussian vectors extend to n dimensions and the sum Z = ni=1 Xi of n Gaussian r.v.s is itself Gaussian with mZ = 2 mi and Var(Z) = σZ2 = σ if the Xi are independent and mZ = i 2 i i 2 = m and σ σ + 2 ρ i i i i i<j ij σi σj if they are not independent (ρij is the Z correlation coefﬁcient between Xi and Xj ).

The multivariate case

119

Example 3.4(a) Consider the random variable Z1 = X1 X2 . If we deﬁne Z2 = X2 then X1 = Z1 /X2 and | det( J)| = 1/|x2 |. Therefore ∞ fZ1 (z1 ) = −∞

1 fX (z1 /x2 , x2 ) dx2 |x2 |

(3.112)

Example 3.4(b) If, on the other hand, we consider the ratio Z1 = X1 /X2 – and, as above, we deﬁne Z2 = X2 – then X1 = Z1 X2 and | det(J)| = |x2 |. Therefore ∞ fZ1 (z1 ) =

|x2 |fX (z1 x2 , x2 ) dx2

(3.113a)

−∞

In addition, if the two original r.v.s are independent with pdfs f1 (x1 ), f2 (x2 ) ∞ fZ1 (z1 ) =

0 x2 f1 (z1 x2 )f2 (x2 ) dx2 −

x2 f1 (z1 x2 )f2 (x2 ) dx2 (3.113b)

−∞

0

So, for instance, if X1 , X2 are independent Gaussian r.v. with m1 = m2 = 0 and Var(X1 ) = σ12 , Var(X2 ) = σ22 then the term at the exponentials in both integrals of eq. (3.113b) can be written as x22 z12 σ22 + σ12 a = −x22 − 2 2 2 b σ 1 σ2 where we deﬁned a = z12 σ22 + σ12 and b = 2σ12 σ22 . Eq. (3.113b) then becomes ⎧∞ ⎫ 0 ⎨ ⎬ 1 x2 exp(−ax22 /b) dx2 − x2 exp(−ax22 /b) dx2 fZ1 (z1 ) = ⎭ 2π σ1 σ2 ⎩ −∞

0

and performing the change of variable t = ax22 /b so that (b/2a) dt = x2 dx2 we get 2 fZ1 (z1 ) = 2π σ1 σ2

b 2a

∞ 0

e−t dt =

σ 1 σ2 2 π(z1 σ22 + σ12 )

(3.114)

where we took into account that the two integrals within braces are equal to twice the integral in dt from 0 to ∞ and we substituted the explicit expressions for a and b to obtain the ﬁnal term on the r.h.s. of (3.114). The

120 Probability theory pdf of eq. (3.114) is a form of the so-called Cauchy distribution. In particular, if X1 , X2 are (independent) standardized r.v.s, then σ1 = σ2 = 1 and fZ1 (z1 ) =

1 π(z12

(3.115)

+ 1)

which is the form of the Cauchy distribution commonly found in the literature.

3.5.1

Numerical descriptors of functions of random variables

In the preceding section we determined how to obtain the probability distribution of a random variable (vector) which is a function of another random variable (vector) when we know the distribution of the original r.v. Depending on the functional relation between the two variables (vectors), this may not always be an easy task. It often happens, however, that the analyst’s interest lies in the numerical descriptors of Z = g(X) rather than in a complete probabilistic description of Z (i.e. fZ or FZ ). Moreover, in most cases one is mainly interested in the ﬁrst and second order moments of Z. These quantities can be obtained – or, more generally, approximated – without going through the determination of fZ or FZ . Starting from the case in which Z is one-dimensional we have already considered (Propositions 2.13 and 2.15; see also eq. (2.35c)) the situation when Z is a linear function of X, that is, Z = aX + b where a, b are two constants. Then E(Z) = aE(X) + b and Var(Z) = a2 Var(X), which, in turn, are special cases of the more general relations

E(Z) = Var(Z) = =

n i=1 n i=1 n i=1

ai E(Xi ) + b a2i Var(Xi ) + 2

ai aj Cov(Xi , Xj )

(3.116)

i<j

a2i Var(Xi ) +

ai aj Cov(Xi , Xj )

ij(i =j)

which occur whenever Z is a linear function of more than one r.v., that is, when Z = ni=1 ai Xi + b. If, in addition, the variables X1 , . . . , Xn are pairwise uncorrelated (or, more strictly, independent), the second of (3.116) becomes Var(Z) = i a2i Var(Xi ). Before turning to the general discussion, consider for instance the frequently encountered non-linear case Z = XY. Then, by the properties of

The multivariate case

121

covariance (Proposition 2.15 or eq. (3.19a)) we have mZ ≡ E(Z) = E(XY) = E(X)E(Y) + Cov(X, Y)

(3.117)

which becomes E(Z) = E(XY) = E(X)E(Y) whenever X, Y are uncorrelated or independent. Moreover if X, Y are independent it is left to the reader to show that the variance of Z is given by Var(Z) = Var(X)Var(Y) + E2 (X)Var(Y) + E2 (Y)Var(X) = σX2 σY2 + m2X σY2 + m2Y σX2

(3.118)

(Hint: start from the deﬁnition Var(Z) = E[(Z − mZ )2 ] and then take into account that X 2 , Y 2 are also independent r.v.s.) Let us now tackle the general problem. We will do so in three steps: in the order (a) a one-dimensional variable function of another one-dimensional variable, (b) a one-dimensional variable function of a random vector and (c) a random vector function of another random vector. Let now Z = g(X) where both X and Z are assumed to be absolutely continuous. If the function g is invertible then we have E(Z) =

z fZ (z) dz =

g(x)fX (x) dx

(3.119)

because z = g(x), dz = g (x) dx (the prime indicates the derivative) and, from eq. (2.71), fZ (z) = fX (x)/g (x) since dg −1 (z)/dz = 1/g (x). However, we can expand g(x) in a Taylor series around mX as z = g(x) = g(mX ) + (x − mX )g (mX ) +

1 (x − mX )2 g (mX ) + · · · 2 (3.120)

and insert this expression in (3.119) to get the approximate relation E(Z) ∼ = g(mX ) +

1 g (mX ) Var(X) 2

(3.121)

because it is easily veriﬁed that the term with the ﬁrst derivative yields zero in the integration. The calculation of the variance is a bit more involved. Similarly to eq. (3.119) we can write Var(Z) =

2

(z − mZ ) fZ (z) dz =

[g(x) − mZ ]2 fX (x) dx

(3.122)

122 Probability theory and use (i) the Taylor expansion (3.120) to approximate g(x) and (ii) eq. (3.121) to approximate mZ = E(Z). After a few passages we arrive at Var(Z) ∼ = [ g (mX )]2 Var(X) +

1 [ g (mX )]2 {M4 − Var2 (X)} 4

+ g (mX )g (mX )M3

(3.123a)

where we denoted by M3 and M4 the third and fourth-order central moments of X, respectively (i.e. M3 = E[(X − mX )3 ] and M4 = E[(X − mX )4 ]; also, using this notation note that Var(X) = M2 ). If the pdf fX (x) is symmetric about the mean, then M3 = 0 and if, in addition, it is Gaussian then (eq. (2.42d)) M4 = 3 M22 = 3 Var2 (X); therefore Var(Z) ∼ = [g (mX )]2 Var(X) +

1 [g (mX )]2 Var2 (X) 2

(3.123b)

For approximation purposes, one may sometimes use mZ = g(mX ) – which is equivalent to interchanging the expectation operator with the functional dependence, that is, E[g(X)] = g[E(X)] – for the mean and σZ2 = [g (mX )]2 σX2 for the standard deviation; however, it should be kept in mind that these relations are exact only in case of a linear relation between X and Z. Let now Z be a function of n random variables X1 , . . . , Xn , that is, Z = g(X1 , . . . , Xn ). In this case the linear approximation is frequently used; in other words one assumes that (i) the mean of the function equals the function of the X-means m1 , . . . , mm and (ii) the variance of the function depends only on the ﬁrst derivatives of g and on the variances σ12 , . . . , σn2 of X1 , . . . , Xn . Although this may seem a rather crude approximation, it generally leads to acceptable result and consequently – besides speciﬁc applications where a higher accuracy is required – linearization is the main technique to deal with the case Z = g(X1 , . . . , Xn ). So, linearizing the function g in a neighbourhood of m1 , . . . , mm we have g(x) = g(m) +

n ∂g (xi − mi ) + · · · ∂xi x=m

(3.124)

i=1

so that inserting this expression in E(Z) =

g(x) fX (x) dx

(3.125)

all the terms with the ﬁrst derivatives go to zero in the integration and mZ = E(Z) ∼ = g(m) = g(m1 , m2 , . . . , mn )

(3.126)

The multivariate case

123

Equation (3.126), in turn, can be used together with (3.124) in the expression Var(Z) =

[g(x) − mZ ]2 fX (x) dx

(3.127)

to arrive at the (approximate) result σZ2 = Var(Z) ∼ =

n

[Di g(m)]2 σi2 +

[Di g(m)][Dj g(m)] Kij (3.128a)

i,j; i =j

i=1

where, for short, we denoted Di g(m) = ∂g/∂xi |x=m . If the variables X1 , . . . , Xn are uncorrelated then Var(Z) ∼ =

n

[Di g(m)]2 σi2

(3.128b)

i=1

which is, nonetheless, an approximation due to the fact that we retained only the ﬁrst-order terms in the Taylor expansion. Introducing the column matrix D whose elements are the ﬁrst-order derivatives of g calculated at x = m, that is, ⎡

⎤ D1 g(m) ⎢D2 g(m)⎥ ⎢ ⎥ D=⎢ ⎥ .. ⎣ ⎦ . Dn g(m) then eq. (3.128a) can be concisely written in matrix form as Var(Z) ∼ = DT KD

(3.128c)

where K is the covariance matrix introduced in eq. (3.32a). If, in addition, the variables X1 , . . . , Xn are uncorrelated then K = diag(σ12 , . . . , σn2 ) and eq. (3.128c) reduces to the sum of squares of eq. (3.128b). A better approximation to E(Z) and Var(Z) than eqs (3.126) and (3.128), respectively, can be obtained by retaining the next term in the Taylor expansion (3.124). This term contains the second-order derivatives of g and can be written as n 1 2 1 2 Di g(m)(xi − mi )2 + Dij g(m)(xi − mi )(xj − mj ) 2 2 i=1

i,j; i =j

124 Probability theory where D2i g(m) = ∂ 2 g/∂x2i |x=m and D2ij g(m) = ∂ 2 g/∂xi ∂xj |x=m . In this approximation we are led to 1 2 1 2 Dij g(m) Kij Di g(m) σi2 + mZ ∼ = g(m) + 2 2

(3.129a)

i,j; i =j

i

or, if the variables are uncorrelated 1 2 Di g(m) σi2 mZ ∼ = g(m) + 2

(3.129b)

i

For the variance we can limit the calculations to the independent (or uncorrelated) case – although eq. (3.128b) will, in general, sufﬁce in this circumstance – and arrive at the rather lengthy relation σZ2 ∼ =

%2 1 $ 2 [Di g(m)]2 σi2 + Di g(m) {M4 (Xi ) − Var2 (Xi )} 4 i i $ % % $ + D2ij g(m) σi2 σj2 [Di g(m)] D2i g(m) M3 (Xi ) + i=j

i

(3.130) where we denoted by M3 (Xi ), M4 (Xi ) the third and fourth-order central moments of the variable Xi , respectively. As an example, we can return to the case Z = XY (X and Y independent) considered above. The reader can check that the approximation (3.128b) does not lead to the correct result (3.118) while, on the other hand, eq. (3.130) does. Finally, we examine now the most general case of m r.v.s Z1 , . . . , Zm which are functions of n r.v.s X1 , . . . , Xn . The situation is as follows Z1 = g1 (X1 , . . . , Xn ) Z2 = g2 (X1 , . . . , Xn ) .. .

(3.131)

Zm = gm (X1 , . . . , Xn ) m the means of Z1 , . . . , Zm , the linear approximation Denoting by m 1, . . . , m immediately yields m k ∼ = gk (m1 , . . . , mn ),

k = 1, 2, . . . , m

(3.132)

of the Z-variables is given by while the covariance matrix K ∼ K = DT KD

(3.133a)

The multivariate case

125

where K is the covariance matrix of the X-variables and we denoted by D the n × m matrix of derivatives ⎤ ⎤ ⎡ ⎡ ∂g1 /∂x1 ∂g2 /∂x1 . . . ∂gm /∂x1 D11 D21 . . . Dm1 ⎢∂g1 /∂x2 ∂g2 /∂x2 . . . ∂gm /∂x2 ⎥ ⎢D12 D22 . . . Dm2 ⎥ ⎥ ⎥ ⎢ ⎢ D=⎢ ⎥ = ⎢ .. .. .. ⎥ .. .. .. .. .. ⎦ ⎣ ⎣ . . ⎦ . . . . . . ∂g1 /∂xn ∂g2 /∂xn . . . ∂gm /∂xn D1n D2n . . . Dmn and it is understood that all derivatives are calculated at the point x = m. So, the (i, j)th element of the matrix is ij = Cov(Zi , Zj ) ∼ K =

Di k Kk l Dj l

k, l

(3.133b)

ji . being a covariance matrix, is clearly symmetric, that is, K ij = K and K, Clearly, eq. (3.133b) could also be directly obtained from the deﬁnition of covariance. In fact, for example, if Z1 = g1 (X1 , X2 ) and Z2 = g2 (X1 , X2 ) we have 12 = Cov(Z1 , Z2 ) = (z1 − m 1 )(z2 − m 2 ) fZ (z) dz K and by a similar line of reasoning as above we can expand both g1 , g2 in a neighborhood of m and use this expansion together with eq. (3.132) to get 12 ∼ K =

∂g1 ∂g2 Kkl ∂xk ∂xl k,l

which, as expected, is the same as eq. (3.133b). As the next example will show, a ﬁnal point worthy of notice is that independence of the X-variables does not, in general, imply independence of the Z-variables. Example 3.5 Let X1 , X2 be two uncorrelated r.v.s with variances K11 = σ12 , K22 = σ22 . Also let Z1 = 2X1 + X2 and Z2 = 5X1 + 3X2 . Then *

2 D KD = 5 T

+"

1 3

σ12

0

0

σ22

#*

2 1

+ " 4 σ12 + σ22 5 = 3 10 σ12 + 3σ22

10 σ12 + 3σ22

#

25 σ12 + 9 σ22

showing that Z1 , Z2 are, as a matter of fact, correlated. Example 3.6 Suppose that the coordinates x, y in a plane can be measured with uncertainties σ1 = 0.2 cm for the x-coordinate and σ2 = 0.4 cm for the y-coordinate. Assume further that the measured x, y values of a point in the

126 Probability theory plane are uncorrelated and they are considered the mean coordinates for that point. Our measurement yields (x, y) = (1, 1); what are the uncertainties in polar coordinates? Now, the functional relations for the problem are , r = x2 + y 2 θ = arctan(y/x) and the derivative matrix is * √ + * 1/√2 x/r −y/r2 = D= y/r x/r2 (x,y)=(1,1) 1/ 2

+

−1/2 1/2

while K = diag(σ12 , σ22 ) = diag(0.04, 0.16). Therefore * 0.100 T ∼ KD = K D = 0.042

+

0.042 0.050

√ and √ the uncertainties we are looking for are σr = 0.1 = 0.32 cm and σθ = 0.05 = 0.22 radians. Consequently, we will express our measurement as x = 1.0 ± 0.2; y = 1.0 ± 0.4 cm in rectangular coordinates and r = 1.41 ± 0.32 cm; θ = π/4 ± 0.22 radians in polar coordinates. Note that the transformation from rectangular to polar coordinates has introduced a positive correlation between r and θ.

3.6

Summary and comments

This chapter continues along the line of Chapter 2 by extending the discussion to the so-called multivariate case, that is, the case in which two, three, . . ., n random variable are considered simultaneously. In this light it is useful to introduce the concept of random vector and – whenever convenient – exploit the brevity and compactness of vector and matrix notation. A n-dimensional random vector X is, in essence, a measurable function from an abstract probability space (W, S, P) to Rn and this implies that each one of its components must be a random variable. In this light, Section 3.2 shows that the familiar concepts of induced probability measure, PDF and pdf (when it exists) can be readily extended to these vector-values functions. A new aspect, which has no counterpart in the one-dimensional case, is considered in Section 3.2.1 where the notion of marginal distribution functions is introduced. These functions have to do with the ‘subvectors’ of a given vector X and it is shown that the joint probability description of X contains implicitly the probabilistic description of each one of its possible ‘subvectors’. In general, however, the reverse statement is not true unless its components are independent. In this case, in fact, a number of important ‘product rules’ hold and one can obtain the joint-PDF (or pdf) of the vector from the PDFs (pdfs) of its components.

The multivariate case

127

Similarly to the one-dimensional case, the moments of a random vector are deﬁned as abstract Lebesgue integrals in the probability space (W, S, P). The most important moments in applications are the ﬁrst- and second-order moments which are given special names. So, in addition to the concepts of mean values and variances of X, the notion of covariance is deﬁned in Section 3.3 and some properties of these numerical descriptors are given. Particularly important in both theory and applications is the notion of uncorrelation of random variables which, broadly speaking, is a weak form of (pairwise) independence. Stochastic independence, in fact, implies uncorrelation but the reverse, in general, is not true. Besides this, Section 3.3 introduces the concept of joint-characteristic function by generalizing the one-dimensional case of Chapter 2; in particular, it is shown that independence implies the validity of a ‘product rule’ also for characteristic functions. Then, in Section 3.3.1 the discussion continues by noting the usefulness of matrix notation and by considering the actual calculations of moments and expectations in practice. In fact, mathematical analysis provides all the necessary results to show that the abstract Lebesgue integrals with respect to the measure P are evaluated as Lebesgue–Sieltjes integrals in Rn ; these, in turn, in most practical cases become either sums or ordinary Lebesgue integrals depending on the type of PDF – that is, FX – induced by the random vector X. Moreover, when the pdf exists the Lebesgue integrals coincide with the familiar Riemann integrals (it should be remembered, however, that Lebesgue integrals have a number of desirable properties which are not satisﬁed by Riemann integrals). Next, Section 3.3.2 is more application-oriented and gives two important examples of multivariate distributions: a discrete one, the so-called multinomial distribution, and a continuous one, the multivariate Gaussian (or normal) distribution. This is done in order to show how the developments considered so far are translated into practice. For its importance in both theory and practice, Sections 3.4 and 3.4.1 return on the subject of conditional probability. Here we extend the notion of conditioning to random variables by also considering, in the continuous case, the possibility of conditioning on events of zero probability. Then, since a conditional probability is a probability measure in its own right, the concepts of conditional PDF and pdf are introduced in the multivariate case and their relation to the joint and marginal functions is also shown. As one might expect, conditional expectations satisfy all the main properties of expectations. However, some additional properties are worthy of mention and these are given in Section 3.4.1 together with further theoretical remarks and examples. Finally, the last two Sections 3.5 and 3.5.1, deal with the probabilistic description of functions of a given random vector X, assuming that some information on X is available. More speciﬁcally – limiting for the most part the discussion to the continuous case – Section 3.5 considers the general problem of obtaining the joint-pdf of a vector Z = g(X); then, in

128 Probability theory order to show practical cases, some examples are given. On the other hand, Section 3.5.1 addresses the problem of obtaining some information on Z without necessarily trying to describe it completely. The task is accomplished by calculating the lowest-order moments – typically means, variances and covariances – of Z only on the basis of the available information on X. In most cases one only arrives at approximate relations because linearization of the function g is often necessary. Nonetheless, this partial information – obtained, in addition, by means of approximate equations – is sufﬁcient and sufﬁciently accurate in a large number of practical situations.

References and further reading [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Brémaud, P., ‘An Introduction to Probabilistic Modeling’, Springer-Verlag, New York (1988). [3] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press, 19th printing (1999). [4] Dall’Aglio, G., ‘Calcolo delle Probabilità’, Zanichelli, Bologna (2000). [5] Friedman, A., ‘Foundations of Modern Analysis’, Dover Publications, New York (1982). [6] Gnedenko, B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [7] Heathcote, C.R., ‘Probability, Elements of the Mathematical Theory’, Dover Publications, New York (2000). [8] Horn, R.A., Johnson, C.R., ‘Matrix Analysis’, Cambridge University Press (1985). [9] Kolmogorov, A.N., Fomin, S.V., ‘Introductory Real Analysis’, Dover, New York (1975). [10] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [11] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000). [12] Pfeiffer, P.E., ‘Concepts of Probability Theory’, 2nd edn., Dover Publications, New York (1978). [13] Rotondi, A., Pedroni, P., Pievatolo, A., ‘Probabilità, Statistica e Simulazione’, Springer-Verlag, Italia, Milano (2001). [14] Biswas, S., ‘Topics in Statistical Methodology’, Wiley Eastern Limited, New Delhi (1991). [15] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997). [16] Thompson, R.S.H.G., ‘Matrices: Their Meaning and Manipulation’, The English Univerities Press Ltd., London (1969). [17] Ventsel, E.S., ‘Teoria delle Probabilità’, Mir Publisher, Moscow (1983).

4

4.1

Convergences, limit theorems and the law of large numbers

Introduction

In most issues where chance plays a part, things seem to behave rather erratically if one looks only at a few instances. On the other hand, this type of behaviour seems to ‘smooth out’ in the long run. In other words, as the number of observed instances – or trials or experiments – increases, a more and more orderly pattern seems to ensue and certain regularities become clearer and clearer. This is what happens, for example, when we toss a coin; after 10 tosses we would not be surprised to have, say, eight heads and two tails but we would surely be if we got 800 heads and 200 tails after 1000 tosses. In fact, in this case we would seriously suspect that the coin is biased. This state of affair would be intriguing but not particularly interesting if it applied only to coins and dice. As a matter of fact, however, a large number of experiences in many ﬁelds of human activities – from birth and death rates to accidents, from measurements in science and technology to the occurrence of hurricanes or earthquakes, just to name a few – behave in a similar manner when measured, tabulated and/or assigned numerical values. The appearance of long-term regularities as the number of trials increases has been known for centuries and goes under the name of ‘law of large numbers’. The great achievement of probability theory is in having established the general conditions under which these regularities can and do occur. We open here a short parenthesis. Returning to the coin example for a moment, it is worth pointing out that the law of large numbers does not justify certain mistaken beliefs such as, say: I tossed a fair coin 15 times and I got 14 heads, the next toss is very likely to result in a head. This is wrong because the process has no memory and the probability of a head is 0.50 for each toss. In other words, the coin has no responsibility whatsoever to ‘make up’ for a past run of many heads in a row. This misinterpretation (unfortunately, a rather common misinterpretation; consider, for example, the habit of betting on ‘late’ numbers in lotteries) of the law of large numbers is due to the fact that one fails to distinguish between a regularity ‘in the ratio sense’ and a regularity in an ‘absolute sense’. The former concept refers to

130 Probability theory the number of heads (or tails) divided by the total number of tosses while the latter refers to the number of heads (or tails) in excess over tails (heads); as the number N of tosses increases, the above ratio tends to stabilize by getting closer and closer to 0.50 while the difference between heads and tails can become rather large (in fact, it generally increases). So, returning to our main discussion, this chapter is intended to provide the mathematical rationale behind the general term ‘law of large numbers’ and since the concept implies a tendency towards something, it is easily guessed that its mathematical formalization entails some kind of limit. The ﬁrst step, therefore, is to consider which kind of limits are involved in the long-term behaviour of experiments governed by chance.

4.2

Weak convergence

In the ﬁnal part of Section 2.4 (Deﬁnition 2.5) we introduced the notion of weak convergence of random variables. This type of convergence is also known in probability theory as ‘convergence in distribution’ or ‘convergence in law’ to mean that the probability law (i.e. the PDF) of Xn converges to a function which is itself a probability law. We recall here some important points: (a) Fn → F[w] – or equivalently Xn → X[D] – means that limn→∞ Fn (x) = F(x) at all points where F(x) is continuous (there is no ambiguity because F(x), being a PDF, is right-continuous). Also, it is not difﬁcult to see that Deﬁnition 2.5 of weak convergence is equivalent to stating that limn→∞ P(Xn ≤ x) = P(X ≤ x) whenever P(X = x) = 0; (b) since weak convergence does not refer directly to the r.v.s Xn and neither it involves directly the probability space on which they are deﬁned (weak convergence is a property of the PDFs and not of the Xn themselves), the concept makes sense even if the Xn are deﬁned on different probability spaces; (c) sequences of discrete r.v.s may converge (weakly) to a continuous r.v.s and conversely. Moreover, the fact that a sequence Xn of absolutely continuous r.v.s with pdfs fn = Fn converges in distribution to an absolutely continuous r.v. X whose pdf is f = F does not imply, in general, that the sequence fn converges to f . It is worth noting, however, that if fn → f pointwise (or even almost everywhere, see Section 4.3), then Xn → X[D]. The extension to random vectors is rather straightforward: if (Xn(1) , Xn(2) , . . . , Xn(m) ) converges weakly to the vector (X (1) , X (2) , . . . , X (m) ) then Xn(i) → X (i) [D] for every i = 1, 2, . . . , m. The reverse in general is not true and weak convergence of every individual component does not imply the vector weak convergence. This result should be hardly surprising; in fact, given F (1) and F (2) – we are considering the two-dimensional case for

Limits, convergences and the law of large numbers

131

F (1) , F (2)

simplicity – there are inﬁnite joint-PDFs for which are the marginal (i) (i) PDFs and therefore Fn → F [w] for i = 1, 2 gives no information on the convergence of FXn . In addition, FXn may not even converge at all. Also – in the light of the deﬁnition of weak convergence – it should be noted that Xn → X[D] does not imply Xn − X → 0[D], as it is customary for ordinary convergence of real variables. A fundamental result on D-convergence is given by Levy’s theorem of Proposition 2.24 which brings into play pointwise convergence of characteristic functions and is often used in probability theory. We use it, for instance, to prove a ﬁrst limit theorem: Proposition 4.1 Let Xn be a sequence of binomial r.v.s with parameters n and p = λ/n, where λ is a positive real number. Then, as n → ∞, Xn converges in distribution to a Poisson r.v. of parameter λ. Before proving this proposition, some preliminary comments on the Poisson distribution are in order. As it is probably known to the reader, we call Poisson r.v. with parameter λ a discrete r.v. X whose pmf is given by pX (x) = e−λ

λx x!

(x = 0, 1, 2, . . .)

(4.1)

and it can be shown that E(X) = Var(X) = λ. In fact, for example, E(X) =

∞ x=0

xe−λ

∞

λx−1 λx e−λ = λ =λ (x − 1)! x!

(4.2a)

x=1

because on the r.h.s. we sum on all the ordinates of the distribution and therefore the sum equals 1. In addition, the CF of the Poisson distribution is easily obtained as ϕ(u) = E(eiuX ) = e−λ

(λeiu )x x

−λ

=e

iu

x!

(4.2b) iu

exp(λe ) = exp[λ(e − 1)]

from which, using eqs (2.47b) and (2.34), it is almost immediate to determine that E(X 2 ) = λ + λ2 and Var(X) = λ. For higher-order moments it may be more convenient to use the recursion relation

d E(X k ) = λ + 1 E(X k−1 ) (4.2c) dλ with the starting assumption E(X 0 ) = 1. Therefore E(X) = λ, E(X 2 ) = λ + λ2 , E(X 3 ) = λ + 3λ2 + λ3 , E(X 4 ) = λ + 7λ2 + 6λ3 + λ4 , etc.

132 Probability theory A ﬁnal remark on the Poisson distribution is as follows: let X, Y be two independent Poisson r.v.s with parameters λ1 , λ2 , respectively. Independence implies (eq. (3.29)) that the CF of the r.v. X + Y is ϕX+Y (u) = {exp[λ1 (eiu − 1)]}{exp[λ2 (eiu − 1)]} = exp[(λ1 + λ2 )(eiu − 1)]

(4.3)

which is the CF of a Poisson r.v. with parameter λ1 + λ2 . This property of reproducing itself by addition of independent variables – possessed also by the Gaussian distribution – is noteworthy and often useful in practice. Moreover, a result by Rajkov shows that the reverse is also true: if the sum of two independent r.v. has a Poisson distribution then each individual r.v. is Poisson distributed. This, we recall (remark in Example 3.3) is true also for Gaussian r.v.s. Now, returning to our main discussion, we know from eq. (2.51) that the CF of the binomial r.v. Xn is given by ϕn (u) = (1 − λ/n + λeiu /n)n . Passing to the limit as n → ∞ we get lim ϕn (u) = lim

n→∞

n→∞

λ(eiu − 1) 1+ n

n = exp[λ(eiu − 1)]

(4.4)

which proves the assertion of Proposition 4.1. On the practical side, this proposition is interpreted by saying that the Poisson distribution – besides being often applicable in its own right – can be used as a valid approximation of the binomial distribution when the probability of ‘success’ p is rather small and n is sufﬁciently large. In fact it should be noted that all the binomial r.v.s Xn have the same mean E(Xn ) = pn = (λ/n)n = λ, thus implying that for large values of n the probability p must be small (incidentally, it is for this reason that the Poisson distribution is often called the distribution of rare events). In this light, as Example 4.1 will show, the parameter λ represents the average number of occurrences of the event under study per measurement unit (of time, length, area, etc., depending on the case). As a general rule of thumb one can use the Poisson distribution to approximate the binomial when either n ≥ 20 and p ≤ 0.05 or when n ≥ 100 and np ≤ 10; this makes calculations much easier because if we are interested in, say, the probability of 9 successes out of n = 1000 trials in a binomial process with p = 0.006 (so that λ = np = 6) it is certainly easier to calculate (69 e−6 )/9! rather than

1000 (0.006)9 (1 − 0.006)1000−9 9

(incidentally, the result of both expressions is 0.0688).

Limits, convergences and the law of large numbers

133

Example 4.1 Two typical cases of Poisson r.v. are as follows. Consider the number of car accidents per month at a given intersection where it is known that, on average, there are 1.7 accidents per month. In this case the month is our measurement unit and the Poisson law can be justiﬁed as follows. Divide a month in n intervals, each of which is so small that at most one accident can occur with a probability p = 0. Then, since it is reasonable to assume that the occurrence of accidents is independent from interval to interval, we are in essence observing a Bernoulli trial where the probability of ‘success’ p is relatively small if n is large. Also we know that λ = np = 1.7 and we can, for instance, obtain the probability of zero accidents in a month as (1.70 e−1.7 )/0! = 0.183. The second example arises from a ballistic problem rather common during II World War. The probability of hitting an airplane in a vulnerable part when shooting with a riﬂe – that is, a ‘success’ – is very low, say, p = 0.001. However, if an entire military unit shoots, say, n = 4000 bullets, one can use the Poisson distribution to determine that the probability of at least two hits x −4 0 −4 1 −4 is (since λ = np = 4) 4000 x=2 4 e /x! = 1−(4 e /0!)−(4 e /1!) = 0.908 which is rather high and has been conﬁrmed in practice. Another important limit theorem – which involves D-convergence and points in the direction of the central limit theorem to be considered in a later section – was ﬁrst partially obtained by deMoivre in the eighteenth century and then completed by Laplace some 60–70 years later. Once again, one considers a sequence of Bernoulli trials and deﬁnes the random variables Xn (n = 1, 2, . . .) which take on the value 0 in case of ‘failure’ or the value 1 in case of ‘success’ (recall that the probability of ‘success’ p does not change from trial to trial). In this light the r.v. Sn = X1 + X2 + · · · + Xn represents the number of successes in n trials and is binomially distributed with mean √ np and standard deviation npq (Example 2.8a). With these assumptions we have the deMoivre–Laplace theorem: Proposition 4.2 Let Sn be the number of successes in a sequence of Bernoulli trials, then

b 1 Sn − np ≤b = √ exp(−z2 /2) dz lim P a < √ n→∞ npq 2π

(4.5)

a

uniformly for all a, b (−∞ ≤ a < b ≤ ∞). The proof is not given here because this proposition is just a particular case of the central limit theorem which will be proven in a later section (Proposition 4.22). Noting that the r.h.s. of eq. (4.5) is P(a ≤ Z < b) where Z is a standard Gaussian r.v., we can state Proposition 4.2 in words by saying that √ the sequence of r.v.s Yn = (Sn − np)/ npq – which, in turn, is obtained by

134 Probability theory ‘standardizing’ the sequence of binomial r.v.s Sn – converges in distribution to a standard Gaussian r.v. This result is also frequently expressed by saying that the r.v. Yn is ‘asymptotically standard normal’ and sometimes written Yn ≈ As−N(0, 1) where N(0, 1) denotes the normal probability distribution with zero mean and unit variance (i.e. the standard Gaussian distribution). In the light of the considerations above, it turns out that – in the limit of large n – the binomial distribution can be approximated either by a Poisson distribution or by a standardized Gaussian. Which one of the two approximations to use depends on the problem at hand; broadly speaking, the Gaussian approximation works well even for moderately large values of n (say n ≥ 20 − 25) as long as p is not too close to 0 or 1. If, on the other hand, p is close to 0 or 1, n must be rather large in order to obtain reasonably good results and in these cases the Poisson approximation is preferred. General rules of thumb are often given in textbooks and one ﬁnds, for example, that √ the Gaussian approximation is appropriate whenever (i) p ± 2 pq/n lies in the interval (0, 1) or (ii) np ≥ 5 if p ≤ 0.5 or nq ≥ 5 if p > 0.5. A third important and useful result considers the asymptotic behaviour of Poisson r.v.s. The CF of a Poisson r.v. X is given by eq. (4.2b);√as a consequence the CF of the standardized Poisson r.v. Y = (X − λ)/ λ is given by √ √ ϕY (u) = exp[−iu λ + λ(eiu/ λ − 1)]

(4.6)

where eq. (4.6) – since Y and X are linearly related – is obtained by using eq. (3.39b). ∞ we can expand the exponential in parenthesis as √ As λ → √ exp(iu/ λ) = 1 + iu/ λ − u2 /2λ + · · · and obtain lim ϕY (u) = exp(−u2 /2)

λ→∞

(4.7)

which, in other words, means that Y ≈ As − N(0, 1). In the light of Propositions 4.1 and 4.2, this last result is hardly unexpected. 4.2.1

A few further remarks on weak convergence

It has been pointed out in the preceding section that weak convergence (or convergence in distribution or in law) concerns the convergence of PDFs and, in general, does not imply the convergence of pmfs or pdfs (when they exist). However, in some cases there is the possibility of establishing ‘local’ limit theorems for these functions. An example is given by the ‘local’ version of the DeMoivre–Laplace theorem (see e.g. [9] or [13]) stating that √ npqBn (m) lim √ =1 n→∞ ( 2π )−1 exp(−x2 /2)

(4.8a)

Limits, convergences and the law of large numbers

135

where

Bn (m) =

n m n m n−m p (1 − p)n−m = p q m m

(4.8b)

,

x = (m − np)/ npq

In words, the result of eq. (4.8a) is expressed by saying that, for any √ given m, the binomial pmf (multiplied by its standard deviation npq) tends to a standardized Gaussian pdf as n gets larger and larger. As a matter of fact, the approximation is rather good even for relatively small values of n. So, for example, if n = 25, p = 0.2 and we are inter√ ested in m = 3, √ then npqB25 (3) = 0.2715 and since x = −1 in this case, we get ( 2π )−1 exp(−x2 /2) = 0.2420. A graphical representation of this local theorem is given in Figures 4.1 (n = 25, p = 0.2) and 4.2 (n = 100, p = 0.2) where one can immediately notice the quality of the approximation: good in the ﬁrst case and excellent in the second case. The reader should check, however, that larger and larger values of n are needed for a good approximation as p gets close to either 0 or 1. As stated in the preceding section, when p is close to either 0 or 1 (say p < 0.1 or p > 0.9) the binomial pdf can be better approximated by a Poisson density. In fact, if we let n → ∞ and p → 0 so that λ = pn is ﬁnite,

0.42 Binomial Std. Gauss.

0.35 0.28 0.21 0.14 0.07 0.00 –2.5

–1.50

–0.50

0.50

1.50 x values

2.50

Figure 4.1 Gaussian approx. to binomial (n = 25, p = 0.2).

3.50

4.5

136 Probability theory 0.42 Binomial Std. Gauss.

0.35 0.28 0.21 0.14 0.07 0.00 –3.50

–2.50

–1.50

–0.50

0.50

1.50

2.50

3.50

x values

Figure 4.2 Gaussian approx. to binomial (n = 100, p = 0.2).

then for any ﬁxed value of m Bn (m) =

n! pm (1 − p)n−m m!(n − m)!

λm λ n−m 1 − m!nm n

m (1 − λ/n)n 1 λ m−1 = nm 1 − ··· 1 − n n m!nm (1 − λ/n)m = n(n − 1) · · · (n − m + 1)

and therefore, since (1 − λ/n)n → exp(−λ) as n → ∞, lim Bn (m) =

n→∞

λm −λ e m!

(4.9)

The approximations considered here, clearly, are not the only ones. So, for example, it may be reasonable to expect that a distribution arising from an experiment of sampling without replacement can be approximated by a distribution of a similar experiment with replacement if the total number of objects N from which the sample is taken is very large. In fact, as N → ∞ and we extract a ﬁnite sample, it no longer matters whether the extraction is done with or without replacement because the probability of ‘success’ is unaffected by the fact that we replace – or do not replace – the extracted item. In mathematical terms these considerations can be expressed by saying that, under certain circumstances, the so-called hypergeometric distribution – which is relative to sampling without replacement – can be approximated by a binomial distribution (see e.g. [18, Section 3.1.3] or [7, Appendix 1,

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Section 17]). However, we do not consider other cases here and, if needed, postpone any further consideration.

4.3

Other types of convergence

Consider a sequence of r.v.s Xn deﬁned on the same probability space (W, S, P). We say that Xn converges in probability to the r.v. X – also deﬁned on (W, S, P) – if for every ε > 0 we have lim P{w ∈ W : |Xn (w) − X(w)| ≥ ε} = 0

n→∞

(4.10a)

and in this case one often writes Xn → X[P] and speaks of P-convergence. It is worth noting that convergence in probability is called ‘convergence in measure’ in mathematical analysis. In words, eq. (4.10a) states that the probability measure of the set where Xn differs from X by more than any prescribed positive number tends to zero as n → ∞. This, we point out, does not assure that all the values |Xn (w) − X(w)| will be smaller than ε for n larger than a certain N, but only that the probability measure of the event (i.e. set) for which |Xn (w) − X(w)| ≥ ε is very small (zero in the limit). Also, it may be noted that eq. (4.10) can be expressed equivalently by writing lim P{w ∈ W : |Xn (w) − X(w)| ≤ ε} = 1

n→∞

(4.10b)

and it is immediate to see that Xn → X[P] if and only if Xn − X → 0[P] (remember that this is not true in general for convergence in distribution). In the case of random vectors the condition (4.10a) – or (4.10b) – must hold for all their components and it is understood that the sequence of vectors Xn and the limit X must have the same dimension. More speciﬁcally, it can be shown that Xn → X[P] if and only if Xn(k) → X (k) [P] for all k (where k is here the index of component; so, for a m-dimensional vector k = 1, 2, . . . , m). We turn our attention now on some important results on convergence in probability starting with the following two propositions: Proposition 4.3 If Xn → X[P] and g : R → R is a continuous function, then g(Xn ) → g(X)[P]. Proposition 4.4(a) Convergence in probability implies convergence in distribution. In fact, we have Fn (x) = P(Xn ≤ x) = P(Xn ≤ x ∩ X > x + ε) + P(Xn ≤ x ∩ X ≤ x + ε) ≤ P(|X − Xn | ≥ ε) + P(X ≤ x + ε) = P(|X − Xn | ≥ ε) + F(x + ε)

138 Probability theory where the inequality comes from two facts: (a) P(Xn ≤ x ∩ X ≤ x + ε) ≤ P(X ≤ x + ε) because of the straightforward inclusion (Xn ≤ x ∩ X ≤ x + ε) ⊆ (X ≤ x + ε), and (b) P(Xn ≤ x ∩ X > x + ε) ≤ P(|X − Xn | ≥ ε) because (Xn ≤ x ∩ X > x + ε) ⊆ (|X − Xn | ≥ ε). This inclusion is less immediate but the l.h.s. event implies x < X − ε and, clearly, Xn ≤ x; consequently Xn < X − ε, which, in turn, is included in the event |X − Xn | ≥ ε. By a similar line of reasoning we get F(x − ε) = P(X ≤ x − ε) = P(X ≤ x − ε ∩ Xn > x) + P(X ≤ x − ε ∩ Xn ≤ x) ≤ P(|X − Xn | ≥ ε) + P(Xn ≤ x) = P(|X − Xn | ≥ ε) + Fn (x) Putting the two pieces together leads to F(x − ε) − P(|X − Xn | ≥ ε) ≤ Fn (x) ≤ P(|X − Xn | ≥ ε) + F(x + ε) and since Xn → X[P] then Fn (x) is bracketed between two quantities that – as ε → 0 – tend to F(x) whenever F is continuous at x. This, in turn, means that Xn → X[D] and the theorem is proven. The reverse statement of Proposition 4.4a is not true in general because – we recall – convergence in distribution can occur for r.v.s deﬁned on different probability spaces, a case in which P-convergence is not even deﬁned. However, when the Xn are deﬁned on the same probability space, a partial converse exists: Proposition 4.4(b) If Xn converges in distribution to a constant c then Xn converges in probability to c. We do not prove the proposition but only point out that: (i) a r.v. which takes on a constant value c with probability one – that is, such that PX (c) = 1 – is not truly random. Its PDF is F(x) = 0 for x < c F(x) = 1 for x ≥ c and often one speaks of ‘degenerate’ or ‘pseudo’ random variable in this case; (ii) when all the Xn and X are deﬁned on the same probability space and X is not a constant, there are special cases in which the converse of Proposition 4.4 may hold (see [11, Chapter 4]). The last result on P-convergence we give here is called Slutsky’s theorem and its proof can be found, for example, in Ref. [1]

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If Xn → X [D] and Yn → c [D] (and therefore Yn →

Proposition 4.5 c [P]), then

(a) Xn + Yn → X + c [D] (b) Xn Yn → cX [D] (c) Xn /Yn → X/c [D] if c = 0. Turning now to another important notion of convergence, we say that the sequence of r.v.s Xn converges almost-surely (some authors say ‘with probability 1’) to X if 3 4 P w ∈ W : lim Xn (w) = X(w) = 1 n→∞

(4.11)

and we will write Xn → X [a.s.] or Xn → X[P − a.s.] if the measure needs to be speciﬁed. Clearly, Xn → X [a.s.] if and only if Xn − X → 0 [a.s.]. Deﬁnition 4.11 implies that the set N of all w where Xn (w) fails to converge to X(w) is such that P(N) = 0 and that, on the other hand, Xn (w) → X(w) for all w ∈ N c where, clearly, P(N c ) = 1. Given a measure P – and a probability is a ﬁnite, non-negative measure – in mathematical analysis one speaks of ‘convergence almost-everywhere’ (a.e.) when condition (4.11) holds; therefore a.s.-convergence is just the probabilistic name given to the notion of a.e.-convergence of advanced calculus. In general, there is no relation between a.e.-convergence and convergence in measure (eq. (4.10)); however, the fact that P is a ﬁnite measure has an important consequence for our purposes: Proposition 4.6 Almost-sure convergence implies convergence in probability (and therefore, by Proposition 4.4, convergence in distribution). This result is a consequence of the following criterion for a.s.-convergence: the sequence Xn converges almost surely to X if and only if for every ε > 0 ⎡ ⎤ ∞ (4.12a) lim P ⎣ {|Xk − X| ≥ ε}⎦ = 0 n→∞

k=n

or, equivalently, ⎡ lim P ⎣

n→∞

∞

⎤ {|Xk − X| < ε}⎦ = 1

(4.12b)

k=n

In fact, if (4.12a) holds then eq. (4.10) follows by virtue of the fact that the probability of a union of events is certainly not less than the probability of each one of the individual events in the union. The proof of the criterion is more involved and is not given here; the interested reader may refer, for

140 Probability theory example, to [16] or [17]. Regarding the converse of Proposition 4.6 – which is not, in general, true – a remark is worthy of notice: it can be shown that if Xn → X[P] then there exists a subsequence Xnk of Xn such that Xnk → X [a.s.] as k → ∞. Proposition 4.7 If Xn → X [a.s.] and g is a continuous function, then g(Xn ) → g(X) [a.s.]. In fact, for every ﬁxed w such that Xn (w) → X(w) then Yn (w) ≡ g(Xn (w)) → g(X(w)) ≡ Y(w) because of the continuity of g. Therefore {w: Xn (w) → X(w)} ⊆ {w: Yn (w) → Y(w)} so that P{w: Yn (w) → Y(w)} ≥ P{w: Xn (w) → X(w)} and the theorem follows. The last comment we make here on a.s.-convergence regards random vectors. As for P-convergence, a sequence of m-dimensional random vectors Xn = (Xn(1) , . . . , Xn(m) ) converges a.s. to the m-dimensional vector X = (X (1) , . . . , X (m) ) if and only if Xn(k) → X (k) [a.s.] for all k = 1, 2, . . . , m. Before turning to the collection of results known as ‘law of large numbers’, we close this section by introducing another type of convergence. A sequence of r.v.s Xn is said to converge to X ‘in the kth mean’ (k = 1, 2, . . .) if lim E(|Xn − X|k ) = lim |Xn − X|k dP = 0 (4.13) n→∞

n→∞

W

and we will write Xn → X [Mk ]. In the above deﬁnition it is assumed that all the Xn and X are such that E(Xnk ) < ∞ and E(X k ) < ∞ because these conditions imply the existence of the expectation in eq. (4.13). In fact, from the inequality |Xn − X|k ≤ 2k (|Xn |k + |X|k ) we can pass to expectations to get E(|Xn − X|k ) ≤ 2k E(|Xn |k ) + 2k E(|X|k ) so that the l.h.s. is ﬁnite whenever the r.h.s. is. Also, it is easy to see that Xn → X [Mk ] if and only if Xn − X → 0 [Mk ]. The most important special cases of (4.13) in applications are k = 1 – the so-called ‘convergence in the mean’ – and k = 2, called ‘convergence in the quadratic mean’. This latter type plays a role in probability when only ‘second-order data’ are available, that is, when the only information is given by the means mn = E(Xn ) and covariances Kij (i, j = 1, . . . , n) and one cannot determine whether the sequence converges in any one of the modes considered before. However, the following result holds: Proposition 4.8 If Xn → X[Mk ] – with k being any one integer – then Xn → X [P] and therefore (Proposition 4.4) Xn → X [D]. In fact, consider Chebyshev’s inequality (eq. (2.36a)) applied to the r.v. Xn − X; for every ε > 0 we have P(|Xn − X| ≥ ε) ≤ E(|Xn − X|k )/ε k and therefore the l.h.s. tends to zero whenever the r.h.s. does. So, in particular, if a sequence converges in the mean or in the quadratic mean then convergence

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in probability and convergence in distribution follow. Furthermore, by virtue of Proposition 2.12, it is immediate to show that convergence in the quadratic mean implies convergence in the mean or, more generally: Proposition 4.9 Convergence in the kth mean implies convergence in the jth mean for all integers j ≤ k. 4.3.1

Additional notes on convergences

In the preceding section we have determined the following relations: (a) a.s.-convergence is stronger than P-convergence which, in turn, is stronger than D-convergence unless the limit is a constant random variable. (b) Mk -convergence (for any one integer k) implies P-convergence and therefore D-convergence. At this point one may ask, for instance, about the relation between Mk and a.s.-convergence. The answer is that, in general, without additional assumptions, there are no relations other than the ones given above. An example is given by the celebrated Lebesgue dominated convergence theorem which, for our purposes, can be stated as follows Proposition 4.10 Let Xn → X [a.s.] or Xn → X [P] and let Y be a r.v. such that E(Y) < ∞ (i.e. with ﬁnite mean) and |Xn (w)| ≤ Y(w) for each n and for almost all w ∈ W. Then Xn → X [M1 ]. (see Ref. [8] or [15]). Note that the expression |Xn (w)| ≤ Y(w) for almost all w ∈ W brings into play the measure P and means that the set N where the inequality does not hold is such that P(N) = 0 (again, this is the ‘almost everywhere’ notion of mathematical analysis). Another important result establishes a relation between D- and a.s.convergence. This is due to Skorohod and, broadly speaking, states that convergence in distribution can be turned into almost sure convergence by appropriately changing probability space. Proposition 4.11 (Skorohod’s theorem) Let Xn and X be r.v.s deﬁned on a probability space (W, S, P) and such that Xn → X [D]. Then, it is possible 5-n and X to construct a probability space (W, S, P) and random variables X such that P(X ≤ x) = P(X ≤ x), P(Xn ≤ x) = P(Xn ≤ x) for n = 1, 2, . . . -n → X - [ˆ (i.e. F(x) = F(x) and Fˆ n (x) = Fn (x) for all n) and X P − a.s.]. We do not prove the theorem here but it is worth noting that, in essence, Proposition 4.11 is due to the fact that any PDF F : R → [0, 1] can be ‘inverted’ to obtain a r.v. deﬁned on the interval U = [0, 1] whose PDF

142 Probability theory 5is F. In this light, it turns out that (W, S, P) = (U, B(U), µ) – where µ is the Lebesgue measure. For more details the interested reader can refer, for example, to [1, 2] or [19]. A third remark of interest is that P-, a.s.- and Mk -convergence can all be established by the well-known Cauchy criterion of mathematical analysis. So, for example, if a sequence Xn satisﬁes the Cauchy criterion in probability, that is, lim P(|Xm − Xn | ≥ ε) = 0

m,n→∞

(4.14)

(which can also be written Xm − Xn → 0 [P] as m, n → ∞), then there exists a r.v. X such that Xn → X [P]. The fact that Xn → X [P] implies eq. (4.14) is clear; therefore it can be said that the Cauchy criterion (4.14) is a necessary and sufﬁcient condition for the sequence Xn to converge (in probability) to a r.v. X deﬁned on the same probability space. Similarly, it can be shown that |Xm − Xn | → 0 [a.s.] implies that there exists X such that Xn → X [a.s.]; consequently, by the same reasoning as above Xn → X [a.s.] if and only if |Xm − Xn | → 0 [a.s.]. By the same token, Xn → X [Mk ] if and only if the Cauchy criterion E(|Xm − Xn |k ) → 0(m, n → ∞) in the kth mean holds. In mathematical terminology, these results can be expressed by saying that the ‘space’ of random variables deﬁned on a probability space (W, S, P) is complete with respect to P, a.s. and Mk convergence. Moreover, if we consider as equal any two r.v.s which are almost everywhere equal (with respect to the measure P) the spaces of r.v.s. with ﬁnite kth order moment (k = 1, 2, . . .) are the so-called Lk spaces of functional analysis. It is well known, in fact, that deﬁning the norm Xk = {E(|X|k )}1/k these are Banach spaces (i.e. complete normed spaces) and, in particular, the space L2 is a Hilbert space. Although it is beyond our scopes, this aspect of probability theory has far-reaching consequences in the light of the fact that the study of Banach and Hilbert spaces is a vast and rich ﬁeld of mathematical analysis in its own right.

4.4

The weak law of large numbers (WLLN)

Broadly speaking, the so-called ‘law of large numbers’ (LLN) deals with the asymptotic behaviour of the arithmetic mean of a sequence of random variables. Since the term ‘asymptotic behaviour’ implies some kind of limit and therefore a notion of convergence, it is customary to distinguish between the ‘weak’ law of large numbers (WLLN) and ‘strong’ law of large numbers (SLLN), where in the former case the convergence is in the probability sense while in the latter almost sure convergence is involved. Clearly, the attributes of ‘weak’ and ‘strong’ are due to the fact that a.s.-convergence is stronger than P-convergence and therefore the SLLN implies the WLLN. In order to cast these ideas in mathematical form, let us consider the WLLN ﬁrst and start with a general result which is a consequence of Chebychev’s

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inequality. For n = 1, 2, . . . consider a√ sequence {Yn } of r.v.s with ﬁnite means E(Yn ) and standard deviations σn = Var(Yn ). Then our ﬁrst statement is: Proposition 4.12 If the numerical sequence of standard deviations is such that σn → 0 as n → ∞, then for every ε > 0 lim P(|Yn − E(Yn )| ≥ ε) = 0

n→∞

(4.15)

By setting b = ε, the proof follows immediately from the ﬁrst of eq. (2.36b). Now, given a sequence of r.v.s Xk deﬁned on a probability space (W, S, P) we can deﬁne, for every n = 1, 2, . . ., the new r.v. Sn = X1 + X2 + . . . + Xn

(4.16)

n with mean E(Sn ) and variance Var(Sn ). (Note that E(Sn ) = k=1 E(Xk ) while, in the general case, eq. (2.35b) gives Var(Sn ) in terms of the variances and covariances of the original variables Xk . If these variables are independent or uncorrelated then Var(Sn ) = nk=1 Var(Xk ).) With these deﬁnitions in mind, the following propositions hold: Proposition 4.13 (Markov’s WLLN)

Sn − E(Sn ) ≥ε =0 lim P n→∞ n

If Var(Sn )/n2 → 0 as n → ∞ then (4.17)

Proposition 4.14(a) (Chebychev’s WLLN) If the variables Xk are independent or uncorrelated and there exists a ﬁnite, positive constant C such that Var(Xk ) < C for all k (in other words, this latter condition can be expressed by saying that the variances Var(Xk ) are ‘uniformly bounded’), then eq. (4.17) holds. The proof of Proposition 4.13 is almost immediate. If we set Yn = Sn /n then, by hypothesis, Var(Yn ) = Var(Sn )/n2 → 0 as n → ∞ and E(Yn ) = E(Sn )/n. In this light, Proposition 4.13 is a consequence of Proposition 4.12. For n Proposition 4.14 we note ﬁrst that Var(Sn ) = k=1 Var(Xk ) < nC, where the equality holds because of independence (or uncorrelation). Consequently, Var(Sn )/n2 < C/n, and since C/n → 0 as n → ∞ the result follows by virtue of Proposition 4.13. At this point, some remarks are in order. First of all, we note that eq. (4.17) can be rewritten equivalently as (Sn − E(Sn ))/n → 0[P] or Sn /n → E(Sn )/n[P], where Sn /n is the arithmetic mean of the r.v.s X1 , X2 , . . . , Xn . So, if the Xk are such that E(Xk ) = µ for all k, then E(Sn ) = nµ and Sn /n → µ [P]

(4.18)

144 Probability theory meaning that for large n the arithmetic mean of n independent r.v.s (each with ﬁnite expectation µ and with uniformly bounded variances) is very likely to be close to µ. This is what happens, for instance, when we repeat a given experiment a large number of times. In this case we ‘sample’ n times a given r.v. X – which is assumed to have ﬁnite mean E(X) = µ and variance Var(X) = σ 2 – so that X1 , X2 , . . . , Xn are independent r.v.s distributed as X. Then, by calculating the arithmetic mean (X1 + X2 + · · · + Xn )/n of our n observations we expect that X 1 + X 2 + · · · + Xn ∼ Sn = =µ n n

(4.19)

We will have more to say about this in future chapters but, for the moment, we note that a typical example in this regard is the measuring process of an unknown physical quantity Q: we make n independent measurements of the quantity, calculate the mean of these observed values and take the result as a good (if n is sufﬁciently large) estimate of the ‘true value’ Q. Note that the assumptions of Proposition 4.14 are satisﬁed because all the Xk have the same distribution as X so that, in particular, E(Xk ) = µ (if the measurements have no systematic error) and Var(Xk ) = σ 2 (and since σ is a ﬁnite number, the variances are uniformly bounded). The relative frequency interpretation of the probability of an event A (recall Section 1.3) is also dependent on the LLN. In fact, by performing n times an experiment in which A can occur, the relative frequency f (A) of A is 1 f (A) = Ik n n

(4.20)

k=1

where Ik is the indicator function of event A in the kth repetition of the experiment. As n gets larger and larger, it is observed that f (A) tends to stabilize in the vicinity of a value – for example, 0.50 in the tossing of a fair coin or, say, 0.03 for the fraction of defective items in the daily production of a given industrial process – which, in turn, is postulated to be the probability of A. In this light, it is clear that we cannot rigorously prove or disprove the existence, in the real world, of such a limiting value because an inﬁnite number of trials is impossible. The best we can do is to build up conﬁdence in our assumptions and check them against real observations; continued success tends to increase our conﬁdence, thus leading us to believe in the adequacy of the postulate. Returning to our main discussion we note that a special case of Proposition 4.14 is given by the celebrated Bernoulli theorem whose basic assumption is that we perform a sequence of Bernoulli trials and p is the probability of success in each trial. If Xk = Ik – the indicator function of a success in the kth trial – the sum Sn is the total number of successes in

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n trials and is binomially distributed with (Example 2.8) E(Sn ) = np and Var(Sn ) = np(1 − p) = npq. Then, Bernoulli’s theorem asserts: Proposition 4.14(b) (Bernoulli’s WLLN)

Sn lim P − p < ε = 1 n→∞ n

With the above assumptions (4.21)

or, equivalently, Sn /n → p[P]. The proof follows from Markov’s theorem (Proposition 4.13) once we note that Var(Sn )/n2 = pq/n → 0 as n → ∞. Now, although the proof of the theorem may seem almost trivial, we must keep in mind that it was the ﬁrst limit theorem to be proved (in the book Ars Conjectandi published in 1713), and therefore Bernoulli did not have the mathematical resources at our disposal. Moreover, since the theorem states that the average number of successes in a long sequence of trials is close to the probability of success on any given trial, its historical importance lies in the fact that this is the ﬁrst step in the direction of removing the restriction of ‘equally likely outcomes’ – necessary in the ‘classical’ notion of probability – in deﬁning the probability of an event. As a consequence, it provides mathematical support to the idea that probabilities can be determined as relative frequencies in a sufﬁciently long sequence of repeated trials. The different forms of the WLLN given so far assume that all the variables Xi have ﬁnite variance. Khintchine’s theorem shows that this is not necessary if the variables are independent and have the same distribution. Proposition 4.15 (Khintchine’s WLLN) If the r.v.s Xk are independent and identically distributed (iid) with ﬁnite ﬁrst moment E(Xk ) = µ then Sn /n → µ [P]. In order to prove the theorem we can use characteristic functions to show that Sn /n → µ[D]. This, by virtue of Proposition 4.4(b), implies convergence in probability. Let ϕ(u) be the common CF of the variables Xk , then we can write the MacLaurin expansion ϕ(u) = ϕ(0) + iuE(Xk ) + · · · = 1 + iuµ + · · · (see Proposition 2.18(a) and the ﬁrst of eq. (2.47b)) where the excluded terms tend to zero as u → 0. Then, if we call ψ(u) = E[exp(iuSn )] the CF of Sn we have E[exp(iuSn /n)] = ψ(u/n) =

n

ϕ(u/n) = {ϕ(u/n)}n = (1 + iuµ/n + · · · )n

k=1

where we used independence in the second equality. Now, as n → ∞, the last expression on the r.h.s. tends to exp(iuµ) which, in turn, is the CF of a pseudo-r.v. µ. Consequently, Sn /n → µ [D] and therefore

146 Probability theory Sn /n → µ[P]. A different proof of this theorem is based on the so-called ‘method of truncation’ and can be found, for example, in [2] or [9]. At this point it could be asked if there is a necessary and sufﬁcient condition for the WLLN to hold. In fact, all the results above provide sufﬁcient conditions and examples can be given of sequences which obey the WLLN but do not verify the assumptions of any one of the theorems above. Such a condition exists and is given in the next theorem due to Kolmogorov. Proposition 4.16 (Kolmogorov’s WLLN) A sequence Xk of r.v.s with ﬁnite expectations E(Xk ) satisﬁes eq. (4.17) – that is, the WLLN – if and only if

2n lim E n→∞ 1 + 2n

6 =0

(4.22)

where = [Sn − E(Sn )]/n. We do not prove this proposition here and the interested reader may refer, for example, to [9]. However, it is worth noting that the theorem requires neither independence nor the existence of ﬁnite second-order moments. Also, since Kolmogorov’s theorem expresses an ‘if and only if’ statement, it can be said that the various conditions of the propositions above are all sufﬁcient conditions for (4.22) to hold, meaning that they imply (but are not implied by) eq. (4.22). In fact, for example, in case of ﬁnite variances we have 2n 1 ≤ 2n = 2 (Sn − E(Sn ))2 2 1 + n n so that taking expectations on both sides it follows that (4.22) holds whenever Markov’s condition on variances (Proposition 4.13) holds.

4.5

The strong law of large numbers (SLLN)

As stated in the preceding section, the type of convergence involved in the different forms of the SLLN is a.s.-convergence, which, in turn, implies P- and D-convergence. Being a stronger statement than the WLLN, the mathematical proofs of the SLLN are generally longer and more intricate than in the weak case; for this reason we will mainly limit ourselves to the results. The reader interested in the proofs of the theorems can ﬁnd them in the references at the end of the chapter. Historically, the ﬁrst statement of SLLN is due to Borel and is somehow a stronger version of Bernoulli’s theorem (Proposition 4.15): Proposition 4.17 (Borel’s SLLN) Let Xk = Ik (k = 1, 2, . . .) be the indicator function of a success in the kth trial in a sequence of independent trials

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and let p be the probability of success in each trial. Then Sn /n → p [a.s.], where, as before, Sn = X1 + X2 + · · · + Xn . Borel’s theorem, in turn, is a special case of the more general result due to Kolmogorov: Proposition 4.18 (Kolmogorov’s SLLN) Let Xk be a sequence of independent r.v.s with ﬁnite variances Var(Xk ) such that ∞ Var(Xk ) k=1

k2

0

P

max |Sk − E(Sk )| ≥ b ≤

1≤k≤n

Var(Sn ) b2

(4.25)

Note that if n = 1 eq. (4.25) reduces to Chebishev’s inequality (2.36b). Proposition 4.21

Borel–Cantelli lemma consists of two parts:

(a) Let (W, S, P) be a probability space and A1 , A2 , . . . be a sequence of events (i.e. An ∈ S for all n = 1, 2, . . .). If ∞ n=1 P(An ) < ∞, then

P lim sup An = 0

(4.26a)

n→∞

∞ where, we recall from Appendix A, lim supn→∞ An = ∩∞ n=1 ∪k=n Ak is itself an event which, by deﬁnition, occurs if and only if inﬁnitely many of the An s occur. (b) If A1 , A2 , . . . are mutually independent and ∞ n=1 P(An ) = ∞, then

P lim sup An = 1 n→∞

(4.26b)

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∪∞ A . k=n k

To prove the ﬁrst part of the lemma, set En = Then E1 ⊃ E2 ⊃ E3 ⊃ · · · is a decreasing sequence of events and the theorem follows from the chain of relations

∞ P lim sup An = P lim En = lim P(En ) ≤ lim P(Ak ) = 0 n→∞

n→∞

n→∞

n→∞

k=n

where we used the deﬁnition of limit of a decreasing sequence of sets (see Appendix A, Section A1) ﬁrst and then the continuity and subadditivity properties of probability. For the second part of the lemma we can write ⎛ P(EC n)

= P⎝

∞

⎞ ⎠ AC k

⎛ = lim P ⎝

k=n

= lim

m→∞

m

k=n

⎞ ⎠ AC k

k=n m

[1 − P(Ak )] ≤ lim

m→∞

⎡

= lim exp ⎣− m→∞

m→∞

m

m

⎤

= lim

m→∞

m

P(AC k)

k=n

exp[−P(Ak )]

k=n

P(Ak )⎦ = 0

k=n

where we used, in the order, De Morgan’s law (i.e. eq. A.6), the independence of the Ak and the inequality 1 + x ≤ ex (which holds for any real number x). Moreover, the last equality holds because ∞ k=n P(Ak ) = ∞. From the relations above, part (b) of the lemma follows from the fact that P(En ) = 1 − P(EC n ) = 1 for each n.

4.6

The central limit theorem

In problems where probabilistic concepts play a part it is often reasonable to assume that the unpredictability may be due to the overall effect of many random factors and that each one of them has only a small inﬂuence on the ﬁnal result. Moreover, these factors – being ascribable to distinct and logically unrelated causes – can be frequently considered as mutually independent. Since our interest lies in the ﬁnal result and not in the individual factors themselves – which, often, are difﬁcult or even impossible to identify – it becomes important to study the existence of limiting probability distributions when an indeﬁnitely large number of independent random effects combine to yield an observable outcome. Needless to say, the ubiquitous Gaussian distribution is one of these limits and the mathematical results formalizing this fact – that is, convergence to a Gaussian distribution – go under the general name of ‘central limit theorem’ (CLT). The various forms of the theorem differ in the assumptions made on the probabilistic nature of the causes affecting the ﬁnal result.

150 Probability theory In order to cast the above ideas in mathematical form, we consider a sequence X1 , X2 , . . . of independent random variables with ﬁnite means E(X1 ) = m1 , E(X2 ) = m2 , . . . and variances Var(X1 ) = σ12 , Var(X2 ) = σ22 , . . .; also, as in the previous sections, we denote by Sn the sum Sn = X1 + X2 + · · · + Xn whose mean and variance are E(Sn ) = nk=1 mk and Var(Sn ) = nk=1 σk2 , respectively. The simplest case is when the variables Xk are iid; then, by calling m ≡ m1 = m2 = · · · the common mean and σ 2 ≡ σ12 = σ22 = · · · the common variance, we have E(Sn ) = nm and Var(Sn ) = nσ 2 . Since these quantities both diverge as n → ∞, it cannot be expected that the sequence Sn can converge in distribution to a random variable with ﬁnite mean and variance (unless, of course, the case m = 0 and σ = 0). However, we can turn our attention to the ‘standardized’ sequence Yn deﬁned by Sn − E(Sn ) Sn − nm Yn = √ = √ σ n Var(Sn )

(4.27)

whose mean and variance are, respectively, 0 and 1. In this light, the ﬁrst form of the CLT – also known as Lindeberg–Levy theorem – is as follows: Proposition 4.22 (Lindeberg–Levy: CLT for iid variables) Let Xk (k = 1, 2, . . .) be iid random variables with ﬁnite mean m and variance σ 2 ; then Yn → Z[D] as n → ∞, where the symbol Z denotes the standardized Gaussian r.v. whose PDF is 1 FZ (x) = √ 2π

x exp(−t 2 /2) dt −∞

The proposition can be proven by recalling Levy’s theorem (Proposition 2.24) and using characteristic functions. In fact, by introducing the iid r.v.s (with zero mean and unit variance) Uj = (Xj − m)/σ for j = 1, 2, . . ., we have Sn − nm 1 Uj =√ √ σ n n n

Yn =

j=1

Now, denoting by ϕ(u) the common CF of the variables Uj , the existence of ﬁnite mean and variance allows one to write the MacLaurin expansion ϕ(u) = ϕ(0) + uϕ (0) +

u2 u2 ϕ (0) + · · · = 1 − + ··· 2 2

where the dots indicate higher order terms that tend to zero more rapidly than u2 as u → 0. Then, since by virtue of independence the CF ψn of Yn is

Limits, convergences and the law of large numbers √ given by ψn (u) = {ϕ(u/ n)}n , we have √

u2 ψn (u) = {ϕ(u/ n)} = 1 − + ··· 2n

151

n

n

so that letting n → ∞ we get limn→∞ ψn (u) = exp(−u2 /2) (the technicality of justifying the fact that we neglect higher order terms can be tackled by passing to natural logarithms; for more details the reader can refer to [19, Chapter VI, Section 7]). This limiting function is precisely the CF of a standardized Gaussian r.v., therefore proving the assertion Yn → Z ≈ N(0, 1)[D] which, more explicitly, can also be expressed as 1 lim P(a < Yn ≤ b) = √ n→∞ 2π

b exp(−x2 /2) dx

(4.28a)

a

for all a, b such that −∞ ≤ a < b ≤ ∞. Equivalently, by taking a = −1 and b = 1 we can also write

9 1 Sn 2 σ lim P − m < √ exp(−x2 /2) dx = n→∞ n π n

(4.28b)

0

which, for large n, can also be interpreted as an estimate on the probability that the arithmetic mean Sn /n (see √ also the following remark (c)) takes values within an interval of length 2σ/ n centered about the mean m. At this point, a few remarks are in order: (a) The DeMoivre–Laplace theorem (Proposition 4.2) is a special case of CLT of Proposition 4.22. In DeMoivre–Laplace case, in fact, the variables Xj are all binomially distributed with mean p and variance pq = p(1 − q). Consequently, the mean and variance of Sn – the number of successes in n independent trials – are np and npq, respectively, so that eq. (4.28) reduces to eq. (4.5). (b) If the Xj are (independent) Poisson r.v.s with parameter λ, then – by virtue of the ‘self-reproducing property’ of Poisson variables pointed out in Section 4.2 – the variable Sn is also Poisson distributed with parameter = nλ and the CF ψn of the variable Yn is obtained by simply substituting in place of λ in eq. (4.6). Then ψn (u) → exp(−u2 /2) as n → ∞, therefore leading to another important special case of Proposition 4.22. (c) In different words, the statement of Proposition 4.22 can be expressed by saying that the variable Sn is asymptotically Gaussian with mean nm

152 Probability theory and variance nσ 2 . This, in turn, implies that the arithmetic mean 1 Sn Xk = n n n

Xn =

(4.29)

k=1

is itself a r.v. which is asymptotically normal with)mean E(X n ) = m, √ variance Var(X n ) = σ 2 /n and standard deviation Var(X n ) = σ/ n. This fact, we will see, often plays an important role in cases where a large number of elements is involved. In particular, it is at the basis of Gauss’ theory of errors where the experimental value of the (unknown) quantity, say Q, is ‘estimated’ by calculating the arithmetic mean of many repeated measurements under the assumption that the errors are iid random variables with zero mean and ﬁnite variance σ 2 . Then – besides relying on the SLLN stating that X n → Q [a.s.] – if n is sufﬁciently large we can also use the Gaussian distribution to make probability statements regarding the accuracy of our result. More about these and other statistical applications is delayed to later chapters. (d) Berry–Esseen inequality: Since for large values of n the standardized Gaussian can be considered as an approximation of the PDF of the variable Yn , the question may arise on how good is this estimate as a function of n. Now, besides the practical fact – also supported by the results of many computer simulations – that the approximation is generally rather good for n ≥ 10, a more deﬁnite answer can be obtained if one has some additional information on the X variables. If, for example, it is known that these variables have a ﬁnite third-order absolute central moment – that is, E(|X − m|3 ) < ∞ – a rather general result is given by Berry–Esseen inequality which states that for all x |Fn (x) − FZ (x)| ≤ C

E(|X − m|3 ) √ σ3 n

(4.30)

where we called Fn (x) = P(Yn ≤ x) the PDF of Yn , FZ (x), as above, is the standardized Gaussian PDF and C is a constant whose current best estimate is C = 0.798 (see Refs [11–14]). Although Lindeberg–Levy theorem is important and often useful, the requirement of iid random variables is too strict to justify all the cases in which the Gaussian approximation seems to apply. In fact, other forms of the CLT show that the assumption of identically distributed variables can be relaxed without precluding the convergence to the Gaussian distribution. Retaining the assumption of independence, a classical result in this direction is Lindeberg’s theorem. We state it without proof and the interested reader can refer, for instance, to [1, 9] or [19].

Limits, convergences and the law of large numbers

153

Proposition 4.23 (Lindeberg’s CLT) Let X1 , X2 , . . . be a sequence of independent random variables with ﬁnite means E(Xk ) = mk and variances Var(Xk ) = σk2 (k = 1, 2, . . .) and let Fk (x) be the PDF of Xk . If, for every ε > 0 (ε enters in the domain of integration, see eq. (4.31b)) the Lindeberg condition 1 n→∞ Var(Sn ) n

lim

(x − mk )2 dFk (x) = 0

(4.31a)

k=1C k

holds, √ then Yn → Z ≈ N(0, 1)[D], that is, the variable Yn = [Sn − E(Sn )]/ Var(Sn ) converges in distribution to a standardized Gaussian r.v. Z. Two remarks on notation: √ n n 2 (i) Clearly E(Sn ) = Var(Sn ) is the k=1 mk , Var(Sn ) = k=1 σk and standard deviation of Sn . In the following, for brevity these last two parameters will often be denoted by Vn2 and Vn , respectively. (ii) The domain of integration Ck in condition (4.31) is the set deﬁned by Ck = {x : |x − mk | ≥ εVn }

(4.31b)

Basically, the Lindeberg condition is an elaborate – and perhaps rather intimidating-looking – way of requiring that the contribution of each individual Xk to the total be small (recall the discussion at the beginning of this section). In fact, since the variable Yn is the sum of n ratios, that is, Sn − E(Sn ) Xk − mk = Vn Vn n

Yn =

k=1

the condition expresses the fact that each individual summand must be uniformly small or, more precisely, that for every ε > 0

|Xk − mk | lim P ≥ε =0 n→∞ Vn

(4.32)

that is, Vn−1 |Xk − mk | → 0[P], which holds whenever eq. (4.31a) holds since 2

ε P(|Xk − mk | ≥ εVn ) = ε

2 Ck

≤

1 Vn2

1 dFk ≤ 2 Vn

n k=1C k

(x − mk )2 dFk Ck

(x − mk )2 dFk → 0

154 Probability theory where the ﬁrst inequality is due to the fact that the domain of integration Ck includes only those x such that |x − mk | ≥ εVn , that is, (x − mk )2 ≥ ε2 Vn2 . Property (4.32) is sometimes called ‘uniform asymptotic negligibility’ (uan). As it should be expected, the iid case of Lindeberg–Levy theorem is just a special case of Proposition 4.23. In fact, if the Xk are iid variables with ﬁnite means m and variances σ 2 , then the sum in (4.31) is simply a sum of n identical terms resulting in 1 σ2

(x − m)2 dF √

{|x−m|≥εσ n}

7 √ 8 which, in turn, must converge to zero because x : |x − mk | ≥ ε n → ∅ as n → ∞. A second special case of Proposition 4.23 occurs when the Xk are uniformly bounded – that is, |Xk | ≤ M for all k – and Vn2 → ∞ as n → ∞. Then

2

ICk (x − mk )2 dFk ≤ (2M)2 P{|x − mk | ≥ εVn }

(x − mk ) dFk = R

Ck

≤

(2M)2 σk2 ε 2 Vn2

where Chebyshev’s inequality (eq. (2.36b)) has been taken into account in the second inequality. From the relations above the Lindeberg condition follows because n 1 (2M)2 (x − mk )2 dFk ≤ 2 2 → 0 2 Vn ε Vn k=1C k

as n → ∞. A third special case of Lindeberg theorem goes under the name of Liapunov’s theorem which can be stated as follows Proposition 4.24(a) (Liapunov’s CLT) Let Xk be a sequence of independent r.v.s with ﬁnite means mk and variances σk2 (k = 1, 2, . . .). If, for some α > 0, 1 Vn2+α

n E |Xk − mk |2+α → 0 k=1

as n → ∞, then Yn → Z[D].

(4.33a)

Limits, convergences and the law of large numbers

155

In fact, Lindeberg’s condition follows owing to the relations 1 1 2 ) dF |x − mk |2+α dFk (x − m ≤ k k 2+α α Vn2 ε Vn k k Ck

≤

Ck

k E(|Xk − mk | ε α Vn2+α

2+α )

where the ﬁrst inequality holds because |x − mk | ≥ εVn . Liapunov’s theorem is sometimes given in a slightly less general form by requiring that ρk = E(|Xk − mk |3 ) < ∞, that is, that all the Xk have ﬁnite third-order central absolute moment. Then Yn → Z[D] if ⎛ ⎞1/3 n 1 ⎝ ⎠ ρk =0 lim n→∞ Vn

(4.33b)

k=1

At this point it is worth noting that eq. (4.31) is a sufﬁcient but not necessary condition for convergence in distribution to Z. This means that there exist sequences of independent r.v.s which converge (weakly) to Z without satisfying Lindeberg’s condition. However, it turns out that for those sequences Xk (of independent r.v.s) such that lim max

n→∞ k≤n

σk2 Vn2

=0

(4.34)

eq. (4.31) is a necessary and sufﬁcient condition for weak convergence to Z. This is expressed in the following proposition Proposition 4.24(b) (Lindeberg–Feller CLT) Let X1 , X2 , . . . be as in Proposition 4.23. Then the Lindeberg condition (4.31) holds if and only if Yn → Z[D] and eq. (4.34) holds. A slightly different version of this theorem replaces eq. (4.34) by the uan condition of eq. (4.32). The interested reader can ﬁnd both the statement and the proof of this theorem in Ref. [1]. 4.6.1

Final remarks

We close this chapter with a few complementary remarks which, although outside our scopes, can be useful to the reader interested in further analysis. The different forms of CLT given above consider D-convergence which, we recall, is a statement on PDFs and, in general, implies nothing on the convergence properties of pmfs or – when they exist – pdfs. However, in

156 Probability theory Section 4.2.1 we mentioned the local deMoivre–Laplace theorem where a sequence of (discrete) Bernoulli pmfs converges to a standardized Gaussian pdf. This is a special case of ‘lattice distributions’ converging to the standardized Gaussian pdf. Without entering into details we only say here that a random variable is said to have a ‘lattice distribution’ if all its values can be expressed in the form a + hk where a, h are two real numbers, h > 0 and k = 0, ±1, ±2,. . .. Bernoulli’s and Poisson’s distributions are just two examples among others. Under the assumption of iid variables with ﬁnite means and variances, a further restriction on h (the requirement of being ‘maximal’) provides a necessary and sufﬁcient condition for the validity of a ‘local’ version of the CLT. The deﬁnition of maximality for h, the theorem itself and its proof can be found in Chapter 8 of [9]. Also, in the same chapter, the following result for continuous variables is given: Proposition 4.25 Let X1 , X2 , . . . be iid variables with ﬁnite means m and variances σ 2 . If, starting from a certain integer n = n0 the variable Yn = √ (σ n)−1 [Sn − n m] has a density fn (x), then 1 fn (x) − √ exp(−x2/2) → 0 2π uniformly for −∞ < x < ∞ if and only if there exists n1 such that fn1 (x) is bounded. A second aspect to consider is whether the Gaussian is the only limiting distribution for sums of independent random variables. The answer to this question is no. In fact, a counterexample has been given in Proposition 4.1 stating that the Poisson distribution is a limiting distribution for binomial r.v.s. Moreover, even in the case of iid variables the requirement of ﬁnite means and variances may not be met. So, in the light of the fact that a so-called Cauchy r.v., whose pdf is f (x) =

1 π(1 + x2 )

(4.35a)

has not a ﬁnite variance, one might ask, for example, if (4.35) could be a limiting distribution or, conversely, what kind of distribution – if any – is the limit of a sequence of independent r.v.s Xk distributed according to (4.35a). Incidentally, we note that (i) the PDF and CF of a Cauchy r.v. are, respectively 1 1 arctan x + π 2 φ(u) = exp(−|u|) F(x) =

(4.35b)

Limits, convergences and the law of large numbers

157

(ii) in (4.35a), failure to converge to the Gaussian distribution is due to the presence of long ‘inverse-power-law’ tails as |x| → ∞. These broad tails, however, do not preclude the existence of a limiting distribution. Limiting problems of the types just mentioned led to the identiﬁcation of the classes of ‘stable’ (or Levy) distributions and of ‘inﬁnitely divisible’ distributions, where the latter class is larger and includes the former. As it should be expected, the Gaussian, the Poisson and the Cauchy distributions are inﬁnitely divisible (the Gaussian and Cauchy distributions, moreover, belong to the class of stable distributions). For the interested reader, more on this topic can be found, for example, in [1, 9, 10, 21]. The third and last remark is on the multi-dimensional CLT for iid random vectors which, in essence, is a straightforward extension of the onedimensional case. In fact, just as the sum of a large number of iid variables is approximately Gaussian under rather wide conditions, similarly the sum of a large number of iid vectors is approximately Gaussian (with the appropriate dimension). In more mathematical terms, we have the following proposition: Proposition 4.26 Let X1 = (X1(1) , . . . , X1(k) ), X2 = (X2(1) , . . . , X2(k) ), . . . be k-dimensional iid random vectors with ﬁnite mean m and covariance matrix K. Denoting by Sn the vector sum

Sn =

n j=1

⎞ ⎛ n n n Xj = ⎝ Xj(1) , Xj(2) , . . . , Xj(k) ⎠ j=1

j=1

(4.36)

j=1

√ then the sequence (Sn − nm)/ n converges weakly to Z, where Z is a k-dimensional Gaussian vector with mean 0 and covariance matrix K.

4.7

Summary and comments

In experiments involving elements of randomness, long-term regularities tend to become clearer and clearer as the number of trials increases and one of the great achievements of probability theory consists in having established the general conditions under which these regularities occur. On mathematical grounds, a tendency towards something implies some kind of limit, although – as is the case in probability – this is not necessarily the familiar limit of elementary calculus. In this light, Sections 4.2 and 4.3 deﬁne a number of different types of convergence by also giving their main individual properties and, when they exist, their mutual relations. Both sections have a subsection – 4.2.1 and 4.3.1, respectively – where additional remarks are made and further details are considered. In essence, the main types of convergences used in probability theory are: convergence in distribution (or weak convergence), convergence in probability, almost-sure convergence and convergence in the kth median

158 Probability theory (k = 1, 2, . . .). Respectively, they are denoted in this text by the symbols D, P, a.s. and Mk convergence and the main mutual relations are as follows: (i) Mk ⇒ Mj (j ≤ k),

(ii) M1 ⇒ P ⇒ D,

(iii) a.s. ⇒ P ⇒ D.

The relation between Mk - and a.s.-convergence is considered in Proposition 4.10 and partial converses of the implications above, when they exist, are also given. With these notions of convergence at our disposal, we then investigate the results classiﬁed under the name LLN, which concern the asymptotic behaviour of the arithmetic mean of a sequence of random variables. In this regard, it is customary to distinguish between weak (WLLN) and strong (SLLN) law of large numbers depending on the type of convergence involved in the mathematical formulation – that is, P- or a.s.–convergence, respectively. The names ‘weak’ and ‘strong’ follow from implication (iii) above and the two laws are considered in Sections 4.4 (WLLN) and 4.5 (SLLN). So, in Section 4.4 we ﬁnd, for instance, Markov’s, Chebychev’s, Khintchine’s and Kolmogorov’s WLLN, the various form differing on the conditions satisﬁed by the sequence of r.v.s involved (e.g. independence, independence and equal probability distributions, ﬁnite variances, etc.). Also, it is shown that Bernoulli’s WLLN – one of the oldest results of probability theory – is a consequence of the more general (and more recent) result due to Markov. It should be noted that most of the above results provide sufﬁcient conditions for the WLLN to hold and only Kolmogorov’s theorem is an ‘if and only if’ statement. Section 4.5 on the SLLN is basically similar to Section 4.4; various forms of SLLN are given and a noteworthy result is expressed by Proposition 4.19 which shows that for iid r.v.s (a frequently encountered case in applications) the existence of a ﬁnite ﬁrst-order moment is a necessary and sufﬁcient condition for the WLLN to hold. In Section 4.5, moreover, we also give two additional results: (a) Borel– Cantelli lemma and (b) Kolmogorov’s inequality. These are two fundamental results of probability theory in general. The main reason why they are included in this section is because they play a key part in the proofs of the theorems on the SLLN, but it must be pointed out that their importance lies well beyond this context. Having established the conditions under which the LLN holds, Section 4.6 turns to one of the most famous results of probability, the so-called CLT which concerns the D-convergence of sequences of (independent) random variables to the normal distribution. We give two forms of this result: Lindeberg–Levy CLT and Lindeberg’s CLT, where this latter result shows that – provided that the contribution of each individual r.v. is ‘small’ – the assumption of identically distributed variables is not necessary. A number of special cases of the theorem are also considered.

Limits, convergences and the law of large numbers

159

The chapter ends with Section 4.6.1 including some additional remarks worthy of mention in their own right. First, some comments are made on ‘local’ convergence theorems, where the term means the convergence of pdfs (or pmfs for discrete r.v.s) to the normal pdf. We recall, in fact, that Dconvergence is a statement on PDFs and does not necessarily imply the convergence of densities. Second, owing to the popularity of the CLT, one might be tempted to think that the normal distribution is the only limiting distribution of sequences of r.v.s. As a matter of fact, it is important to point out that it is not so because there exists a whole class of limiting distributions and the normal is just a member of this class. The subject, however, is outside the scope of the book and references are given for the interested reader. The third and ﬁnal remark is an explicit statement of the multi-dimensional CLT for iid random vectors.

References [1] Ash, R.B., Doléans-Dade, C., ‘Probability and Measure Theory’, Harcourt Academic Press, San Diego (2000). [2] Boccara, N., ‘Probabilités’, Ellipses – Collection Mathématiques Pour l’Ingénieur (1995). [3] Brémaud, P., ‘An Introduction to Probabilistic Modeling’, Springer-Verlag, New York (1988). [4] Breiman, L., ‘Probability’, SIAM – Society for Industrial and Applied Mathematics, Philadelphia (1992). [5] Cramer, H., ‘Mathematical Methods of Statistics’, Princeton Landmarks in Mathematics, Princeton University Press,19th printing (1999). [6] Dall’Aglio, G., ‘Calcolo delle Probabilità’, Zanichelli, Bologna (2000). [7] Duncan, A.J., ‘Quality Control and Industrial Statistics’, 5th edn., Irwin, Homewood, Illinois. [8] Friedman, A., ‘Foundations of Modern Analysis’, Dover Publications, New York (1982). [9] Gnedenko, B.V., ‘Teoria della Probabilità’, Editori Riuniti, Roma (1987). [10] Gnedenko, B.V., Kolmogorov, A.N., ‘Limit Distributions for Sums of Independent Random Variables’, Addison-Wesley, Reading MA (1954). [11] Heathcote, C.R., ‘Probability: Elements of the Mathematical Theory’, Dover Publications, NewYork (2000). [12] Jacod J., Protter, P., ‘Probability Essentials’, 2nd edn., Springer-Verlag, Berlin (2003). [13] Karr, R.A., ‘Probability’, Springer Texts in Statistics, Springer-Verlag, New York (1993). [14] Klimov, G., ‘Probability Theory and Mathematical Statistics’, Mir Publishers, Moscow (1986). [15] McDonald, J.N., Weiss, N.A., ‘A Course in Real Analysis’, Academic Press, San Diego (1999). [16] Monti, C.M., Pierobon, G., ‘Teoria della Probabilità’, Decibel editrice, Padova (2000).

160 Probability theory [17] Pfeiffer, P.E., ‘Concepts of Probability Theory’, 2nd edn., Dover Publications, New York (1978). [18] Biswas, S., ‘Topics in Statistical Methodology’, Wiley Eastern Limited, New Delhi (1991). [19] Taylor, J.C., ‘An Introduction to Measure and Probability’, Springer-Verlag, New York (1997). [20] Ventsel, E.S., ‘Teoria delle Probabilità’, Mir Publisher, Moscow (1983). [21] Wolfgang, P., Baschnagel, J., ‘Stochastic Processes, from Physics to Finance’, Springer-Verlag, Berlin (1999).

Part II

Mathematical statistics

5

5.1

Statistics: preliminary ideas and basic notions

Introduction

With little doubt, the theory of probability considered in the previous chapters is an elegant and consistent mathematical construction worthy of study in its own right. However, since it all started out from the need to obtain answers and/or make predictions on a number of practical problems, it is reasonable to expect that the abstract objects and propositions of the theory must either have their counterparts in the physical world or express relations between real-world entities. As far as our present knowledge goes, real-world phenomena are tested by observation and experiment and these activities, in turn, produce a set – or sets – of data. With the hope to understand the phenomena under investigation – or at least of some of their main features – we ‘manipulate’ these data in order to extract the useful information. In experiments where elements of randomness play a part, the manipulation process is the realm of ‘Statistics’ which, therefore, is a discipline closely related to probability theory although, in solving speciﬁc problems, it uses techniques and methods of its own. Broadly speaking, the main purposes of statistics are classiﬁed under three headings: description, analysis and prediction. In most cases, clearly, the distinction is not sharp and these classes are introduced mainly as a matter of convenience. The point is that, in general, the individual data are not important in themselves but they are considered as a means to an end: the measure of a certain physical property of interest, the test of a hypothesis or the prediction of future occurrences under given conditions. Whatever the ﬁnal objectives of the experiment, statistical methods are techniques of ‘inductive inference’ in which a particular set (or sets) of data – the so-called ‘realization of the sample’ – is used to draw inferences of general nature on a ‘population’ under study. This process is intrinsically different and must be distinguished from ‘deductive inference’ where conclusions based on partial information are always correct, provided that the original information is correct. For example, in basic geometry the examination of particular cases leads to the deduction that the sum of the angles

164 Mathematical statistics of a triangle equals 180 degrees, a conclusion which is always correct within the framework of plane geometry. By contrast, inductive inferences drawn from incomplete information may be wrong even if the original information is not. In the ﬁeld of statistics, this possibility is often related to the (frequently overlooked) process of data collection on one hand – it is evident that insufﬁcient or biased data and/or the failure to consider an important inﬂuencing factor in the experiment may lead to incorrect conclusions – and, on the other hand, to the fact that in general we can only make probabilistic statements and/or predictions. By their own nature, in fact, statements or predictions of this kind always leave a ‘margin of error’ even if the data have been properly collected. To face this problem, it is necessary to design the experiment in such a way as to reduce this uncertainty to values which may be considered acceptable for the situation at hand. Statistics itself, of course, provides methods and guidelines to accomplish this task but the analyst’s insight of the problem is, in this regard, often invaluable. Last but not least, it should always be kept in mind that an essential part of any statistical analysis consists in a quantitative statement on the ‘goodness’ of our inferences, conclusions and/or results. A ﬁnal remark is not out of place in these introductory notes. It is a word of caution taken from Mandel’s excellent book [20]. In the light of the fact that statistical results are often stated in mathematical language, Mandel observes that ‘the mathematical mode of expression has both advantages and disadvantages. Among its virtues are a large degree of objectivity, precision and clarity. Its greatest disadvantage lies in its ability to hide some very inadequate experimentation behind a brilliant facade’. In this regard, it is surely worth having a look at Huff’s fully enjoyable booklet [15].

5.2

The statistical model and some notes on sampling

As explained in Chapter 4, the mathematical models of probability are based on the notion of probability space (W, S, P), where W is a non-empty set, S a σ -algebra of subsets of W (the ‘events’) and P is a probability function deﬁned on S. Moreover, an important point is that one generally considers – more or less implicitly – P to be fully deﬁned. In practice, however, P is seldom known fully and there exists some degree of uncertainty attached to it. Depending on the problem, the degree of uncertainty may vary from a situation of complete indeterminacy – where P could be any probability function that can be deﬁned on S – to cases of partial indeterminacy in which P is known to belong to a given class but we lack some information which, were it available, would specify P completely. In general terms, the goal of statistics is to reduce the uncertainty in order to gain information and/or make predictions on the phenomena under investigation. This task, as observed in the introduction, is accomplished by using and ‘manipulating’ the data collected in experiment(s).

Preliminary ideas and notions

165

In more mathematical terms, the general idea is as follows. We perform an experiment consisting of n trials by assuming that the result xi (i = 1, 2, . . . , n) of the ith trial is associated to a random variable Xi . By so doing, we obtain a set of n observations (x1 , x2 , . . . , xn ) – the so-called realization of the sample – associated to the set of r.v.s (X1 , X2 , . . . , Xn ) which, in turn, is called a random sample (of size n). Both quantities can be considered as n-dimensional vectors and denoted by x and X, respectively. Also, we call sample space the set of all possible values of X and this, depending on X, may be the whole Rn or part of it (if X is continuous) or may consist in a ﬁnite or countable number of points of Rn (if X is discrete). With the generally implicit assumption that there exists a collection of subsets of forming a σ -algebra – which is always the case in practice – one deﬁnes (, ) as the statistical model of the experiment, where here denotes the class of possible candidates of probability functions pertaining to the sample X. Clearly, one of the elements of will be the (totally or partially unknown) ‘true’ probability function PX . Now, referring back for a moment to Chapters 2 and 3, we recall that a random vector X on (W, S, P) deﬁnes a PDF FX which, in turn, completely determines both the ‘induced’ probability PX and the original probability P. Any degree of uncertainty on P, therefore, will be reﬂected on FX (or, as appropriate, on the pmf pX or pdf fX , when they exist) so that, equivalently, we can say that our statistical model is deﬁned by (, ), where is a class of PDFs such that FX (x) = PX (X1 ≤ x1 , . . . , Xn ≤ xn ) ∈ . A particular but rather common situation occurs when the experiment consists in n independent repetitions of the same trial (e.g. tossing a coin, rolling a die, measuring n times a physical quantity under similar conditions, etc.). In this case the components of the sample X1 , X2 , . . . , Xn are iid random variables, that is, they are mutually independent and are all distributed like some r.v. X so that FXi (xi ) = FX (xi ) for all i = 1, 2, . . . , n. The variable X is often called the parent random variable and the set RX of all possible values of X is called the population; also, with this terminology, one can call X a ‘sample (of size n) from the population RX ’. So, depending on the problem at hand, there are two possibilities (1) the PDF FX is totally unknown (and therefore, a priori, may include any PDF), or (2) the general type of FX is known – or assumed to be known – but we lack information on a certain parameter θ whose ‘true’ value may vary within a certain set (note that, in general, θ may be a scalar – then ⊆ R – or a k-dimensional vector (k = 2, 3, . . .), and then ⊆ Rk ). In case (1) our interest may be (1a) to draw inferences on the type of PDF underlying the phenomena under study or (1b) to draw inferences which do not depend on the speciﬁc distribution of the population from which

166 Mathematical statistics the sample is taken. Statistical techniques that are – totally or partially – insensitive to the type of distribution and can be applied ignoring this aspect are called non-parametric or distribution-free methods. In case (2) we speak of parametric model and the class is of the form = {F(x; θ ) : θ ∈ }

(5.1)

where, for every ﬁxed θ ∈ , F(x; θ ) is a well-deﬁned PDF (the semicolon between x and θ denotes that F is a function of x with θ as a parameter, and not a function of two variables). Clearly, one denotes by Pθ the probability function associated to F(x; θ ). In most applications, parametric models are either discrete of absolutely continuous, depending on the type of PDFs in . It is evident that in these cases the model can be speciﬁed by means of a class of pmfs or pdfs, respectively. Example 5.1 (Parametric models) Two examples may be of help to clarify the theoretical discussion above. (i) Consider the experiment of tossing n times a coin whose bias, if any, is unknown. Assuming that we call a head a ‘success’, the natural parent r.v. X associated with the experiment assigns the value 1 to a success and 0 to a failure (tail). Since the experiment is a Bernoulli scheme, the distribution of X will clearly be a binomial pmf (eq. (2.41a)) whose probability of success, however, is not known. Then, our statistical model consists of two sets: , which includes any possible sequence (of n elements) of 1s and 0s, and which includes all the pmfs of the type n x p(x; θ ) = θ (1 − θ )n−x x

(5.2)

where x denotes the number of successes (x = 0, 1, . . . , n), and θ ∈ = [0, 1] because the probability of success can be any number between 0 and 1 (0 and 1 representing the case of totally biased coin). Performing the experiment once leads to a realization of the sample – that is, one element of – which form our experimental data. Statistics, using these data, provides methods of evaluating – estimating is the correct term – the unknown parameter θ, that is, to make inferences on how much biased is the coin. (ii) From previous information it is known that the length of the daily output – say, 5000 pieces – of a machine designed to cut metal rods in pieces of nominal length = 1.00 m, follows a Gaussian distribution. The mean and variance of the distribution, however, are unknown. In this case X is the length of a rod and consists of the density functions f (x; θ1 , θ2 ) =

−1 % √ $ 2π θ2 exp − (x − θ1 )2 /2θ22

(5.3)

Preliminary ideas and notions

167

where, in principle, θ1 ∈ 1 = (−∞, ∞) and θ2 ∈ 2 = (−∞, ∞) but of course – being θ1 , θ2 the mean and standard deviation of the process – the choice can in reality be restricted to much smaller intervals of possible values. By selecting n, say n = 50, pieces of a daily production and by accurately measuring their lengths we can estimate the two parameters, thus drawing inferences on the population (the lengths of the 5000 daily pieces). The realization of the sample are our experimental data, that is, the 50 numbers (x1 , x2 , . . . , x50 ) resulting from the measurement. Although the type of parametrization is often suggested by the problem, it should be noted that it is not unique. In fact, if h : → is a one-to-one function, the model (5.1) can be equivalently written as = {F(x; ψ) : ψ ∈ }

(5.4)

where = {ψ : ψ = h(θ ), θ ∈ } and the choice between (5.1) and (5.2) is generally a matter of convenience. One word of caution, however, is in order: some inferences are not invariant under a change of parametrization, meaning in other words that there are statistical techniques which are affected by the choice of parametrization. This point will be considered in due time if and whenever needed in the course of future discussions. It is worth at this point to pay some attention to the process by which we collect our data, that is, the so-called procedure of sampling. Its importance lies in the fact that inappropriate sampling may lead to wrong conclusions because our inferences cannot be any better than the data from which they originate. If the desired information is not implicitly contained in the data, it will never come out – no matter how sophisticated the statistical technique we adopt. Moreover, if needed, a good set of data can be analysed more that once by using different techniques, while a poor set of data is either hopeless or leads to conclusions which are too vague to be of any practical use. At the planning stage, therefore – after the goal of the experiment has been clearly stated – there are a certain number of questions that need an answer. Two of these, the most intuitive, are: ‘how do we select the sample?’ and ‘of what size?’. In regard to the ﬁrst question the basic prerequisite is that the sample must be drawn at random. A strict deﬁnition of what exactly constitutes a ‘random sample’ is rather difﬁcult to give but, luckily, it is often easier to spot signs of the contrary and decide that a given procedure should be discarded because of non-randomness. The main idea, clearly, is to avoid any source of bias and make our sample, as it is often heard, ‘representative’ of the population under study. In other words, we must adopt a sampling method that will give every element of the population an equal chance of being drawn. In this light, the two simplest sampling schemes are called ‘simple sampling (with replacement)’ and ‘sampling without replacement’. In both cases the sampling procedure is very much like drawing random tickets from an urn: in

168 Mathematical statistics the ﬁrst case we draw a ticket, note the value inscribed, and replace the ticket in the urn while in the second case we do not replace the ticket before the next drawing. It is worth noting that the main difference is that sampling without replacement, in general, cannot be considered as a repetition of a random experiment under uniform conditions because the composition of the population changes from one drawing to another. However, if (i) the population is inﬁnite or (ii) very large and/or our sample consists in a small fraction of it (as a rule of thumb, at most 5%), the two schemes are essentially the same because removal of a few items does not signiﬁcantly alter – case (ii) – the composition of the population or – case (i) – does not alter it at all. The two sampling schemes mentioned above are widely used although, clearly, they are not the only ones and sometimes more elaborate techniques are used in speciﬁc applications. For our purposes, we will generally assume the case of simple sampling, unless otherwise explicitly stated in the course of future discussions. In regard to random samples, a ﬁnal point worthy of mention is that, for ﬁnite populations, the use of (widely available) tables of random numbers is very common among statisticians. The members of the population are associated with the set of random numbers or some subset thereof; then a sample is taken from this set – for example, by blindly putting a pencil down on the table and picking n numbers in that section of the table – and the corresponding items of the population are selected. In case of sampling without replacement we must disregard any number that has already appeared. The number n is the size of the sample which, as noted above, is one of the main points to consider at the planning stage because its value – directly or indirectly – affects the quality and accuracy of our conclusions. However, since the role of the sample size will become clearer as we proceed, further considerations will be made in due time.

5.3

Sample characteristics

It often happens that an experiment consists in performing n independent repetitions of a trial to which a one-dimensional parent r.v. X, with PDF FX , is attached. Then, the sample is the sequence of iid r.v.s X1 , . . . , Xn and the realization of the sample will be is a sequence x1 , x2 , . . . , xn of n observed values of X. Recalling from Section 2.3.2 the numerical descriptors of a r.v. – that is, mean, variance, moments, central moments, etc. – we can deﬁne the sample, or statistical, counterparts of these quantities. So, the ordinary (i.e. non-central) sample moment and sample central moment of order k (k = 1, 2, . . .) are 1 k Xi n

(5.5)

1 (Xi − A1 )k n

(5.6)

n

Ak =

i=1 n

Ck =

i=1

Preliminary ideas and notions

169

respectively, where A1 is the sample mean and C2 is the sample variance. The speciﬁc values of these quantities obtained from a realization of the sample x1 , x2 , . . . , xn will be denoted by the corresponding lowercase letters, that is, ak and ck , respectively. So, for instance, if we repeat the experiment (i.e. other n trials) a second time – thus obtaining a new set x1 , x2 , . . . , xn of observed values – we will have, in general ak = ak and ck = ck (k = 1, 2, . . .). At this point, in order not to get lost in symbols, a few comments on notation are necessary: (a) the sample characteristics are denoted by Ak and Ck to distinguish them from their population (or theoretical) counterparts E(X k ) – with the mean E(X) = µ as a special case – and E[(X − µ)k ] – with the variance E[(X−µ)2 ] = σ 2 (or Var(X)) as a special case. The parent r.v. X and the sample size n to which Ak and Ck refer are often clear from the context and therefore will be generally omitted unless necessary either to avoid confusion or to make a point. (b) Since some population characteristics are given special Greek symbols – for example, µ, σ 2 and the standard deviation σ – it is customary to indicate their sample counterparts by the corresponding uppercase italic letters, that is, M, S2 and S, respectively. So, in the light of eqs (5.5) and (5.6) we have S2 = C2 = n−1

M = A1 ;

, (Xi − M)2 and, clearly, S = S2 . i

(c) Italic lowercase letters, m, s2 and s, denote the speciﬁc realization of the sample characteristic obtained as a result of the experiment, that is: m = n−1

i

xi ; s2 = c2 = n−1

(xi − m)2 and s =

, s2

i

(d) Greek letters will be often used for higher-order population characteristics. So, αk and µk will denote respectively the (population) ordinary and central moments of order k. In this light, clearly, α1 = µ and µ2 = σ 2 but for these lower-order moments the notation µ and σ 2 (or Var(X)) will generally be preferred. The main difference to be borne in mind is that the population characteristics are ﬁxed (though sometimes unknown) constants while the sample characteristics are conceived as random variables whose realizations are obtained by actually performing the experiment. More generally, since Ak and Ck are just special cases of (measurable) functions of X1 , . . . , Xn , the above considerations apply to any (measurable) function G(X1 , . . . , Xn ) of the sample. Any function of this type which contains no unknown parameters is often called a statistic. So, for instance, the Ck deﬁned in (5.6) are statistics while the quantities n−1 i (Xi − µ)k are not if µ is unknown.

170 Mathematical statistics Returning to the main discussion, a ﬁrst observation to be made is that the relations between theoretical moments given in previous chapters still hold true for their sample counterparts and for their realizations as well. Then, for example, by appropriately changing the symbols, eq. (2.34) becomes S2 = A2 − A21 = A2 − M2

(5.7)

or, more generally, the relation (2.33) between central and ordinary moments is Ck =

k (−1)j k! j M Ak−j j!(k − j)!

(5.8)

j=0

and similar equations hold between ak and ck . Moreover, conceiving the sample characteristics as random variables implies that they will have a probability distribution in their own right which, as should be expected, will be determined by FX . Whatever these distributions may be, the consequence is that it makes sense to speak, for instance, of the mean, variance and, in general, of moments of the sampling moments. Let us start by considering the mean E(M) of the sample mean M. Since E(Xi ) = µ for all i = 1, . . . , n, the properties of expectation give 1 1 E(M) = E Xi = E(Xi ) = µ n n i

(5.9)

i

The variance of M, in turn, can be obtained by using eq. (3.116) and the independence of the Xi . Therefore µ2 (M) = Var(M) =

1 1 σ2 Var(X ) = n Var(X) = i n n2 n2

(5.10)

i

√ and the standard deviation is σM = σ/ n. Similarly, it is left to the reader to show that the third- and fourth-order central moments of M are given by % µ $ 3 µ3 (M) ≡ E (M − µ)3 = 2 n % µ $ 3(n − 1) 4 4 µ4 (M) ≡ E (M − µ)4 = 3 + σ n n3

(5.11a)

and so on, with more tedious calculations, for the ﬁfth, sixth order, etc. It is useful, however, to know the order of magnitude of the leading term in

Preliminary ideas and notions

171

these central moments of M; for even and odd moments we have µ2m (M) = O(n−m ), µ2m−1 (M) = O(n−m ),

m = 1, 2, . . .

(5.11b)

respectively, where the symbol O(q) is well known from analysis and means ‘of the same order of magnitude’ of the quantity q in parenthesis. Equation (5.11b) can be checked by looking at (5.10) and (5.11a). Next, turning variance S2 , eq. (5.7) gives our2 attention2 to the sample 2 −1 2 2 E(S ) = n E i Xi − E(M ) = E(X ) − E(M ). Then, using eq. (2.34) for both r.v.s X and M we get its mean as

2 E(S2 ) = σ 2 + µ2 − σM − µ2 = σ 2 −

n−1 2 σ2 = σ n n

(5.12)

a bit more involved. DeﬁningYi = The calculation of the variance of S2 is ¯ 2 , where Y¯ = n−1 Xi − µ(i = 1, 2, . . . , n) we get S2 = n−1 i (Yi − Y) j Yj . Then we can write ⎡ 2 ⎤ 1 1 1 ¯ 2= ⎣ S2 = (Yi − Y) Yi2 − Yi ⎦ n n n i i i ⎤ ⎡ 1 1 2 2 = ⎣ Yi2 − Yi − Yi Yj⎦ n n n i

=

i

n−1 n2

i

i<j

2 Yi2 − 2 Yi Yj n i<j

Squaring this quantity and taking its expectation gives E[(S2 )2 ] =

2

n−1 n2 ⎡⎛

⎡ 2 ⎤ E⎣ Yi2 ⎦ i

⎞2 ⎤ ⎤ ⎡ 4 ⎢⎝ 4(n − 1) ⎥ + 4E⎣ Yi Y j ⎠ ⎦ − E⎣ Yr2 Yi Yj ⎦ n n4 r i<j

i<j

172 Mathematical statistics Now, taking independence into account plus the fact that E(Yi ) = 0 for all i, the last term on the r.h.s. is zero while the ﬁrst and second term lead to ⎞ ⎛

n − 1 2 ⎝ 4 Yi + 2 Yi2 Yj2 ⎠ E n4 i⎞ i<j ⎛ 4 ⎝ Yi2 Yj2 ⎠ E n4

i<j

respectively. Therefore ⎞ 2 2+4 2(n − 1) (n − 1) E[(S2 )2 ] = E ⎝ Yi4 + Yi2 Yj2 ⎠ n4 n4 ⎛

i

=

(n − 1)2 n3

µ4 +

i<j

(n − 1)2

+2

n3

(5.13)

(n − 1)σ 4

where the last equality holds because E(Yi4 ) = µ4 , E(Yi2 ) = σ 2 and there are n(n − 1)/2 combinations of n r.v.s taken two at a time. Finally, since Var(S2 ) = E[(S2 )2 ] − E2 (S2 ), we use eqs (5.13) and (5.12) to get µ2 (S2 ) = Var(S2 ) =

(n − 1)2 n3

µ4 −

n−3 4 σ n−1

(5.14a)

So, in particular, if the original population is normal then the mean and variance of S2 are given, respectively, by eq. (5.12) and by Var(S2 ) =

2(n − 1) 4 σ n2

(5.14b)

where this last result follows from (5.14a) by taking eq. (2.42d) into account. With rather cumbersome calculations one could then go on to obtain µ3 (S2 ), µ4 (S2 ), etc. We do not do it here but limit ourselves to two further results worthy of mention: the ﬁrst concerns the mean and variance of the sample moments Ak and their covariances. It is rather easy to determine E(Ak ) = αk

% α − α2 $ Var(Ak ) ≡ E (Ak − αk )2 = 2k n k $ % α −α α Cov(Ak Al ) ≡ E (Ak − αk )(Al − αl ) = k+l n k l

(5.15)

where the ﬁrst two equations are in agreement with the special cases (5.9) and (5.10) when one notes that A1 = M, α1 = µ and α2 = σ 2 + µ2 . For the

Preliminary ideas and notions

173

order of magnitude of even and odd central moments of the Ak we have % $ µ2m (Ak ) ≡ E (Ak − ak )2m = O(n−m ), % $ µ2m−1 (Ak ) ≡ E (Ak − ak )2m−1 = O(n−m ),

m = 1, 2, . . .

(5.16)

and it is easily seen that eq. (5.11b) are the special case k = 1 of (5.16). The second result gives the covariance between the sample mean and the sample variance; it is left to the reader to show that *

+ n−1 2 n−1 E (M − µ) S2 − σ µ3 = E[(M − µ)S2 ] = n n2

(5.17)

which implies that for any symmetric distribution M and S2 are uncorrelated. In fact, as it is probably known to the reader, µ3 is a measure of skewness – or asymmetry or lopsidedness – of the distribution so that µ3 = 0 for any symmetric distribution. More speciﬁcally, the (adimensional) coefﬁcient of skewness γ1 is often used, where by deﬁnition γ1 =

µ3 3/2 µ2

=

µ3 σ3

(5.18)

With the above results at hand, one can determine the mean and variance of a number of (well-behaved) functions of sample moments by using the approximations given in Section 3.5.1. So, for example, if k, l ≥ 1 are any two integers and g(Ak , Al ) is a twice differentiable function in some neighbourhood of (αk , αl ), then eq. (3.126) gives E[g(Ak , Al ) ∼ = g(αk , αl )

(5.19a)

while eq. (3.128a) leads to Var[g(Ak , Al )] ∼ =

∂g ∂Ak

+2

2

Var(Ak ) +

∂g ∂Al

∂g ∂g Cov(Ak Al ) ∂Ak ∂Al

2 Var(Al ) (5.19b)

where it is understood that all derivatives are calculated at the point (αk , αl ). Note, in particular, that eqs (5.19a) and (5.19b) can be used to approximate the mean and variance of the sample central moment Ck which, as shown by eq. (5.8), is a polynomial in Ak , Ak−1 , . . . , A1 . Example 5.2(a) Consider the mean and variance of C2 = A2 − A21 . From eq. (5.19a) we get E(C2 ) = α2 − µ2 = σ 2 , which is the leading term in the

174 Mathematical statistics exact result (5.12). On the other hand, eq. (5.19b) yields Var(C2 ) ∼ =

µ4 − µ22 α4 − α22 + 8µ2 α2 − 4µ4 − 4µα3 = n n

which, as it should be expected, is the leading term of eq. (5.14) (the second equality is obtained by taking into account the relations between ordinary and central moments). It is evident that the method leading to eqs (5.19a) and (5.19b) is essentially of analytical nature and, as such, it applies to all cases in which the assumptions of the relevant theorems are satisﬁed. These assumptions, in general, have do with the behaviour of the function g and, for this point, the reader is referred to books of mathematical analysis. Example 5.2(b) As a second example, we calculate√the variance of the socalled (sample) ‘coefﬁcient of variation’ V = S/M = C2 /A1 , provided that this quantity is bounded. We have 1 ∂V = √ ∂C2 2µ µ2 √ µ2 ∂V =− 2 ∂A1 µ so that using eq. (5.19b) and retaining only the leading terms in Var(C2 ), Var(A1 ) and Cov(C2 A1 ) – see eqs (5.14), (5.10) and (5.17), respectively – we get Var(V) ∼ =

µ2 (µ4 − µ22 ) − 4µ3 µ2 µ + 4µ32 4nµ2 µ4

(5.20)

By similar calculations one could obtain, for instance, the approximate mean and variance of the sample counterpart of the coefﬁcient of skewness (5.18). 5.3.1

Asymptotic behaviour of sample characteristics

The considerations of the preceding section readily extend to the multidimensional case and the reader is invited to work out the details. Here we turn our attention to another issue: the asymptotic behaviour of sample characteristics as n tends to inﬁnity or, in practical applications, for large samples. Starting with the sample mean M, we can use Markov’s WLLN (Proposition 4.13) to determine that M → E(M)[P]. In fact, since the Xi are iid r.v.s and Sn = X1 + · · · + Xn = nM, then Var(Sn ) = n2 Var(M) = nσ 2

Preliminary ideas and notions

175

)/n2

and Var(Sn → 0 as n → ∞, showing that the assumptions of the theorem are all satisﬁed. Then, by virtue of eq. (5.9) we have M → µ[P]. Actually, by recalling the various form of SLLN given in Section 4.5, one can state the stronger result M → µ[a.s.]. More generally, as a consequence of Chebyshev’s inequality (see also Proposition 4.12) and the ﬁrst of eq. (5.15) we have Ak (n) → E(Ak ) = αk [P] and, even more, by virtue of Proposition 4.19 Ak (n) → αk [a.s.] whenever αk is ﬁnite. Note that here the n in parenthesis stresses the fact that the moments Ak depend on the sample size. Clearly, similar statements are valid for the sample central moments and for any sample characteristic which is a continuous function of a ﬁnite number of the Ak . These convergence properties, in turn, imply that for large values of n the quantities calculated using the data of the experiment can be regarded as ‘estimates’ of the corresponding population characteristics. However, according to certain criteria used to evaluate the quality of the approximation, we will see in later sections that these may not always be the ‘best’ estimates one can ﬁnd. A second aspect to consider is the fact that the quantity nAk = i Xik is a sum of n independent variables – the Xik – which are independent by virtue of Proposition 3.3 and all have the same distribution. As a consequence, it follows that Proposition 5.1(a)

As n → ∞, the standardized variable

√ nA − nαk n(A − αk ) = ) k ) k n(α2k − αk2 ) α2k − αk2 tends in distribution to the standard Gaussian r.v. In fact, since eq. (5.15) imply E(Xik ) = αk and Var(Xik ) = α2k − αk2 for all i = 1, . . . , n, the result follows from Lindeberg–Levy CLT (Proposition 4.22). Also, note that Proposition 5.1 can be stated in different words by saying that Ak is asymptotically normal with mean αk and variance (α2k − αk2 )/n so that, in particular, the sample mean M is asymptotically normal with mean µ and variance σ 2 /n (see also remark (c) after the proof of Proposition 4.22). In this regard, moreover, when sampling from a normal population we have the special result: Proposition 5.1(b) If the parent r.v. X is normal with mean µ and variance σ 2 , M is exactly normal with mean µ and variance σ 2 /n.

176 Mathematical statistics The proof is immediate if we turn to CFs and note that ϕM (u) = E{exp[iu(X1 + · · · + Xn )/n]} = E[iuX1 /n] · · · E[iuXn /n] = [ϕX (u/n)]n Then, since ϕX has the form given in (2.52), ϕM is the CF of a Gaussian r.v. with mean µ and variance σ 2 /n. Continuing along the above line of reasoning, we can use Proposition 4.26 (the multi-dimensional CLT) to show that Proposition 5.2 The joint distribution of any ﬁnite number of sample moments is itself asymptotically normal. In fact, considering the two-dimensional case for simplicity, let r, s ≥ 1 be any two integers; the vector n(Ar , As )T can be written as r r r X Ar X1 Xn n = i is = + · · · + s As X Xns X i i 1 where Xi = (Xir , Xis )T are n iid two-dimensional vectors such that for all i = 1, . . . , n we have the mean E(Xi ) = (αr , αs )T and the covariance matrix

Var(Xir ) K= Cov(Xis Xir )

Cov(Xir Xis ) α2r − αr2 = s Var(Xi ) αr+s − αr αs

αr+s − αr αs α2s − αs2

where the proof of the relation Cov(Xir Xis ) = αr+s −αr α√ s is immediate. Then, Proposition 4.26 states that, as n → ∞, the vector n(Ar − αr , As − αs ) tends in distribution to a Gaussian two-dimensional vector with mean 0 and covariance matrix K. The extension to a higher dimensional case is straightforward. Another important result is as follows: Proposition 5.3 Let g(x, y) be a twice differentiable function in some neigh√ bourhood of (αr , αs ). Then, as n → ∞, the r.v. n[g(Ar , As )−g(αr , αs )] tends in distribution to a normal variable with zero mean and variance DT KD, where D is the vector

∂g/∂Ar D= ∂g/∂As and it is understood that all derivatives are calculated at the point (αr , αs ). In order to sketch the proof, set cr = ∂g/∂Ar and cs = ∂g/∂As . Since g(Ar , As ) − g(αr , αs ) = cr (Ar − αr ) + cs (As − αs ) + · · ·

Preliminary ideas and notions 177 √ the variable n[g(Ar , As )−g(αr , αs )] is a sum of two r.v.s which, by virtue of Proposition 5.2, tend in distribution to a normal r.v. with zero mean, variances cr2 (α2r − αr ) and cs2 (α2s − αs ) and covariance cr cs (αr+s − αr αs ). Then, Proposition 5.3 follows from the fact that the sum of two dependent normal r.v.s A, B with means a, b and variances σA , σB is itself normal with mean a + b and variance σA2 + σB2 + 2Cov(A, B) (see eq. (3.60)). Also note that one can equivalently state the theorem by saying that g(Ar , As ) is asymptotically normal with mean g(αr , αs ) and variance n−1 DT KD where n−1 K is the covariance matrix of the sample moments Ar , As . The extension to cases where g is a function of more than two moments is immediate and, by appropriately deﬁning D, the matrix notation DT KD still applies. Example 5.3 The sample central moments Ck are functions of A1 , A2 , . . . , Ak and therefore Proposition 5.3 includes them as special cases. In fact, any Ck is asymptotically normal with mean µk and variance 1 µ2k − 2kµk−1 µk+1 − µ2k + k2 µ2 µ2k−1 n

(5.21)

where eq. (5.21) can be obtained starting from eq. (5.8) and noting that the central moments do not depend on where we take the origin. Therefore, there is no loss of generality in assuming the origin at the population mean – that is, setting µ = 0 – so that αk = µk and all derivatives ∂Ck /∂Aj are zero except ∂Ck /∂Ak = 1 and ∂Ck /∂A1 = −kµk−1 . Then, since n−1 DT KD = Var(Ak ) + k2 µ2k−1 Var(A1 ) − 2kµk−1 Cov(Ak A1 ) the desired result follows by taking eq. (5.15) into account. So, for instance, the asymptotic variance of C2 is n−1 (µ4 − µ22 ) which, as expected, coincides with the leading term of eq. (5.14a). Returning to our main discussion, it is worth pointing out that the considerations above do not imply that asymptotic normality – although rather common – is a general rule. In order to give an example of sample characteristics which show a different behaviour in the limit of n → ∞ , we must ﬁrst introduce the notion of ‘order statistics’. Suppose, for simplicity, that we are sampling from a continuous population; each realization of the sample x1 , . . . , xn can be arranged in increasing order x(1) ≤ x(2) ≤ · · · ≤ x(n) where, clearly, x(1) = min(x1 , . . . , xn ) and x(n) = max(x1 , . . . , xn ). Then, letting X(k) , k = 1, . . . , n, denote the r.v. that has the value x(k) for each realization of the sample, we deﬁne a new sequence X(1) , X(2) , . . . , X(n) of random variables satisfying X(1) ≤ · · · ≤ X(n) . This new sequence is called the ordered series of the sample and X(k) , in turn, is called the kth order statistic where, in particular, X(1) and X(n) are the extreme values of the sample. A ﬁrst observation is that order statistics are

178 Mathematical statistics not independent because information on one r.v. of the series provides information on other r.v.s: in fact, for example, if X(k) ≥ x then we know that X(k+1) , . . . , X(n) ≥ x. A second observation is that sampling from an absolutely continuous populations prevents the possibility of two or more order statistics being equal since the probability of, say, X(k) = X(k+1) is zero, thus justifying the expression ‘for simplicity’ at the beginning of this paragraph. The PDF of X(k) can be obtained by noting that the event X(k) ≤ x occurs whenever at least k out of the n independent r.v.s X1 , . . . , Xn are ≤ x. Each one of these events has probability F(x) – where F(x) is the PDF of the parent r.v. X and f (x) = F (x) is its pdf. Therefore we have a binomial PDF given by F(k) (x) =

n n j=k

j

[F(x)]j [1 − F(x)]n−j

(5.22a)

which is absolutely continuous if F(x) is. Taking the derivative with respect to x leads to the pdf of X(k) , that is, F(k)

=

n n j=1

j

jF

j−1

(1 − F)

n−j

n n f− (n − j + 1)F j (1 − F)n−j−1 f j−1 j=k+1

+ n * n n n j− (n − j + 1) = kF k−1 (1 − F)n−k f + j j−1 k j=k+1

× F j−1 (1 − F)n−j f and since the term within square brackets is zero we get n f(k) (x) = k[F(x)]k−1 [1 − F(x)]n−k f (x) k

(5.22b)

where it should be noted that in the extreme cases k = 1 and k = n, respectively, eq. (5.22a) reduces to eqs (2.73a) and (2.72a) while (5.22b) agrees with eqs (2.73b) and (2.72b). In order to investigate the behaviour of order statistics as n → ∞ we must distinguish between mid-terms and extremal terms of the ordered series. We call mid-terms the elements whose index is of the form k = [pn] where p is any ﬁxed number 0 < p < 1 and the notation [a] indicates the integer value of the number a. So, in these cases k depends on n and k/n → p as n → ∞. On the other hand, we call extremal terms the elements of the series whose ordinal index is considered ﬁxed throughout the limiting process and has either the form k = r or k = n − s + 1, where r, s are any two ﬁxed integers ≥1. Note that k = n − s + 1 always indicates the sth element from the top, irrespective of the sample size n which, in fact, is assumed to increase indeﬁnitely.

Preliminary ideas and notions

179

Without entering into the details of the calculation, it can be shown (see, for example, [3] or [19]) that the mid-terms are asymptotically normal. However, the extremal terms are not. In fact, for instance, consider X(r) and deﬁne the new variable γ = nF(x), denoting by g(r) (γ ) its pdf. Since we must have f(r) (x) dx = g(r) (γ ) dγ , we get from (5.22b) g(r) (γ ) =

γ n−r γ n−r r n γ r−1 r n γ r 1− 1− = n r n n γ r n n (5.23)

and – being 0 ≤ γ /n ≤ 1 – we can use the limit (4.9) to obtain lim g(r) (γ ) =

n→∞

γ r−1 −γ γ r−1 −γ e = e (r − 1)! (r)

(5.24)

where in the second expression we used the well-known ‘gamma function’ deﬁned in Appendix C. Similarly, if for the sth statistic from the top X(n−s+1) one deﬁnes g = n(1 − F(x)) it is easy to determine that g(s) (γ ) is again given by (5.23), the only difference being the index s in place of r. Therefore limn→∞ g(s) (γ ) = {γ s−1 / (s)}e−γ . The above limiting functions are gamma distributions – (γ ; 1, r) and (γ ; 1, s), respectively – which, however, represent the limit of a function of the relevant order statistics and not of the order statistics themselves. When F(x) is given, it may sometimes be possible to obtain the explicit inverse relation, x = F −1 (γ /n) or x = F −1 (1 − γ /n) as appropriate, but these cases are rather rare. It is worth noting, nonetheless, that considerable work has been done in this direction and it has been found that the limiting distributions of (appropriately standardized) extreme statistics are only of three types, often denoted as Types I, II and III or EV1, EV2 and EV3 – where EV is the acronym for extreme values. Convergence to type I, II or III depends essentially on the ‘tail’ of the underlying distribution F(x) and the rate of convergence is generally rather slow. For more details on this rich and interesting topic the interested reader can refer to [5, 10, 11, 22]. We close this section with two additional comments relevant to the above discussion. First, the sample counterpart of the p-quantile ζp – which is deﬁned implicitly by the equation F(ζp ) = p(0 < p < 1) – is a mid-term order statistic and therefore it is asymptotically normal. It can be shown that its mean and variance are, respectively, ζp and p(1 − p) nf 2 (ζp )

(5.25)

In particular, since ζ1/2 deﬁnes the median of the population, its sample counterpart – X[n/2]+1 if n is odd or any value between X[n/2] and X[n/2]+1 if n is even – is asymptotically normal with mean ζ1/2 and variance {4nf 2 (ζ1/2 )}−1 .

180 Mathematical statistics If, moreover, the parent r.v. is normal with parameters µ, σ 2 , then the sample median is asymptotically normal with parameters µ, (2n)−1 π σ 2 . The second comment refers to the extremal variables X(r) and X(n−s+1) ; if one considers their joint distribution it can be shown that they are asymptotically independent. Both results cited in these comments can be found in [3] or [19].

5.4

Point estimation

As stated at the beginning of this chapter, experimental data are a means to an end: to draw inferences on a population when, for whatever reason, it is not possible to examine the population in its entirety. Clearly, the type of inference – and therefore the desired ﬁnal information – depends on the problem at hand. Nonetheless, some classes of problems are frequently encountered in practice and speciﬁc statistical methods have been devised to address them. Here we consider the parametric model of eq. (5.1) with the aim of ‘estimating’ one or more unknown population parameters. This is one of the typical inference problems and we can choose to give our estimate in one of two distinct forms: (i) by assigning a speciﬁc value to the unknown parameter or (ii) by specifying an interval which – with a given level of conﬁdence – includes the ‘true’ value of the parameter. One speaks of ‘point estimation’ in case (i) and of ‘interval estimation’ in case (ii) and it is understood that (i) and (ii) refer to each one of the unknown parameters when these are more than one. Point estimation is the subject of this and the following sections (Section 5.5 included). Given a sample X1 , . . . , Xn we have considered in the previous sections a number of sample characteristics: each one has the form of a function T(X) = T(X1 , . . . , Xn ) and is a random variable which takes on the value t = T(x) = T(x1 , . . . , xn ) after the experiment has been performed and we have obtained the realization x1 , . . . , xn . If, moreover, T(X) contains no unknown quantities it is generally called a statistic. Intuitively, one would think of estimating an unknown population parameter by using the corresponding statistic so that, for instance, we could use M and S2 as estimators of the population mean and variance µ and σ 2 , respectively. As reasonable as this may sound, things are not always so clear-cut because other statistics can be used for the same purpose and, a priori, there seems to be no reason why M and S2 should be preferred. In order to motivate our choice even in more complex cases, we must ﬁrst try to evaluate the ‘goodness’ of estimators. Let us call θ the unknown parameter to be estimated and let T(X) – or, often, Tn or simply T, implicitly implying the dependence on the sample size – be the statistic used to estimate it. A ﬁrst desirable property for T is that E(T) = θ

for all θ ∈

(5.26)

Preliminary ideas and notions

181

which, in words, is phrased by saying that T is an unbiased estimator (often we will write UE for short) of θ. Note that, strictly, one should write Eθ (T) = θ because the Lebesgue–Stieltjies integral deﬁning the expectation is an integral in dF(x; θ ). This fact, however, is often tacitly assumed for parametric models of the type (5.1). Deﬁning the bias of Tn as b = E(Tn ) − θ it is obvious that Tn is unbiased whenever b = 0. Note that eq. (5.26) does not imply t = θ for every realization of the sample; in fact some realizations will give t − θ > 0 and some others will result in t − θ < 0, however, on average, eq. (5.26) guarantees that there is no systematic error in the evaluation of θ. Also, if g is an arbitrary non-linear function, eq. (5.26) does not imply E[g(Tn )] = √ g(θ ), this meaning, for example, that if Tn is an UE of σ 2 not necessarily Tn is an UE of the standard deviation σ . Besides the bias, another measure of ‘distance’ from the true value is the mean square error (of Tn ), deﬁned as Mse(Tn ) = E[(Tn − θ )2 ]

(5.27a)

which, using the identity T − θ = (T − E(T)) + (E(T) − θ ) = (T − E(T)) + b, can be expressed as Mse(T) = Var(T) + b2

(5.27b)

where E[(T −E(T))2 ] = Var(T) by deﬁnition. Equation (5.27b), in addition, shows that the mean square error of an UE coincides with its variance. So, between any two estimators, say T, T , of the same parameter θ, it seems logical to prefer T if Mse(T) < Mse(T ) for all θ ∈ . If, as it is often the case, we limit our choice to the class of UEs – let us denote this class by u(θ) – the ‘best’ estimator will be the one with minimum variance for all θ ∈ . This minimum-variance-unbiased-estimator (MVUE) T¯ is often called an efﬁcient (or optimum) estimator of θ and satisﬁes the condition ¯ = min {Var(T)} Var(T) T∈u(θ )

for all θ ∈ .

(5.28)

although the concepts are sometimes distinguished because the estimator with minimum variance among all possible estimators (of a given parameter) may not be unbiased. Clearly, one can also speak of relative efﬁciency and compare two estimators on the basis of the ratio of their variances by saying that – given T, T ∈ u(θ) – T is more efﬁcient than T if Var(T) < Var(T ) for all θ ∈ . In this regard, however, it should be noted that it may happen that Var(T) < Var(T ) for some values of θ but Var(T ) < Var(T) for other values of θ. Since the inequality must hold uniformly in θ – that is, for all θ ∈ – and θ is unknown, no efﬁciency comparison can be made in these cases. The same

182 Mathematical statistics consideration applies to efﬁcient estimators and (5.28) may hold for, say, T1 for some θ and T2 for some other θ. Then, efﬁciency is not enough to compare estimators. Within the class u(θ ), the following results hold: Proposition 5.4 If T1 , T2 ∈ u(θ ) are two efﬁcient estimators, then T1 = T2 where the equality T1 = T2 is understood in a probability sense, that is, if T1 and T2 satisfy eq. (5.28) then Pθ (X ∈ {x : T1 (x) = T2 (x)}) = 0 for all θ ∈ . In other words, an efﬁcient estimator, when it exists, is unique. The following proposition, on the other hand, states that efﬁciency is linear: Proposition 5.5 If T1 , T2 , respectively, are efﬁcient estimators of θ1 , θ2 , then a1 T1 + a2 T2 is an efﬁcient estimator of a1 θ1 + a2 θ2 for all a1 , a2 ∈ R. Both proofs can be found in Ref. [19]. A ﬁnal remark is in order: in some cases an UE may not exist or, in other cases, a slightly biased estimator Tb can be preferred to an unbiased one T if Mse(Tb ) < Mse(T) = Var(T) for all θ ∈ . Other desirable properties of estimators consider their behaviour as n → ∞ and not, as above, by regarding the sample size as ﬁxed. These properties are called asymptotic and one says, for instance, that an estimator T is asymptotically unbiased if lim E(Tn ) = θ

n→∞

(5.29)

or equivalently limn→∞ bn = 0, where we write bn because the bias generally depends on the sample size. Clearly, an UE is asymptotically unbiased while the reverse, however, is not true in general. Another asymptotic property is as follows: an estimator Tn of θ is consistent if limn→∞ P(|Tn − θ| < ε) = 1 for all ε > 0, that is, if (see Section 4.3) Tn → θ[P]

(5.30)

Some authors speak of weakly consistent estimator in this case and use the adjective ‘strong’ if Tn → θ [a.s.] or, sometimes, if Tn → θ [M2 ]. In any case (see Propositions 4.6 and 4.8) strong consistency implies weak consistency and, in most cases, the deﬁnition of ‘consistent’ is understood in the sense of eq. (5.30). A useful sufﬁcient condition to determine consistency is given by Proposition 5.6 Tn is a consistent estimator if (a) it is asymptotically unbiased and (b) limn→∞ Var(Tn ) = 0. In fact, if (a) and (b) hold then eqs (5.27b) and (5.27a) imply limn→∞ E[(Tn − θ )2 ] = 0, that is, Tn → θ[M2 ] and therefore Tn → θ[P]. It is evident that

Preliminary ideas and notions

183

(b) only must hold if Tn is unbiased. Note also that requirements (a) and (b) are not necessary; in fact it can be shown that there are consistent estimators whose variance is not ﬁnite. Owing to the properties of P-convergence we have also: Proposition 5.7 If Tn is a consistent estimator of θ and g is a continuous function, then g(Tn ) is a consistent estimator of g(θ ). All the deﬁnitions and considerations above extend readily to the case of more than one unknown parameter θ1 , θ2 , . . . , θk (k > 1) which, as noted in Section 5.2, can be considered as a k-dimensional vector. To end this section, a ﬁnal word of caution on asymptotic properties of estimators is not out of place. In practical cases, these properties provide valid criteria of judgement for large samples but lose their meaning for small samples. Unfortunately, the notions of ‘small’ or ‘large’ samples often depend on the problem at hand and cannot be made more precise without considering speciﬁc cases. A general rule of thumb requires n > 30 in order to be able to speak of ‘large samples’; caution, however, must be exercised because the exceptions to this ‘rule’ are not rare. Example 5.4(a) Equation (5.9) and the ﬁrst of (5.15) show that the statistics M and Ak are UEs of the population parameters µ and αk , respectively. Equation (5.12), however, shows that S2 is a biased estimator of σ 2 , the bias being b = −σ 2 /n. Since b → 0 as n → ∞, S2 is an asymptotically unbiased estimator of σ 2 (also, it is consistent because it satisﬁes the requirements of Proposition 5.6). For ﬁnite samples, nonetheless, the bias can be removed by considering the estimator S¯ 2 =

n 1 S2 (Xi − M)2 = n−1 n−1 n

(5.31a)

i=1

which satisﬁes E(S¯ 2 ) = σ 2 (note that some authors use the name ‘sample variance’ to denote the statistic S¯ 2 ). Also, for a normal population we have from eqs (5.14b) and (5.31a) Var(S¯ 2 ) =

2σ 4 n−1

(5.31b)

The procedure of bias removal shown above can be generalized to all cases in which E(T) = c+dθ – where c, d are two known constants – by deﬁning T¯ = (T − c)/d. Then, the statistic T¯ is an UE of θ. Another example of this type is the statistic C3 as an estimator of µ3 because E(C3 ) = n−2 (n − 1)(n − 2)µ3 (the reader is invited to check this result).

184 Mathematical statistics In cases where the population mean µ is known the quantity 1 Sˆ 2 = (Xi − µ)2 n n

(5.32)

i=1

is a statistic in its own right. It is left to the reader to show that (i) Sˆ 2 is an UE of σ 2 and (ii) Var(Sˆ 2 ) = n−1 (µ4 − σ 4 ) = 2n−1 σ 4 where the last equality is due to the fact that, for a normal r.v., µ4 = 3σ 4 (see eq. (2.44d)). Example 5.4(b) From the considerations above it is evident that, in general, there exist many unbiased estimators of a given parameter. As a further example of this, it is easy to show that any linear combination Tˆ = ni=1 ci Xi such that c1 +c2 +· · ·+cn = 1 is an UE of µ. Turning to its variance, however, ˆ = σ 2 c2 and since we have Var(T) i i i

ci2

=

i

1 ci − n

2 +

1 n

it follows that the minimum value of the sum i ci2 occurs when ci = 1/n for all i = 1, . . . , n. Consequently, the sample mean M is the most efﬁcient ˆ If, in particular, the sample comes among all estimators (of µ) of the form T. from a normal population N(µ, σ 2 ), we noted at the end of the preceding section that the sample median – let us denote it by Z – is asymptotically normal with parameters µ and (2n)−1 π σ 2 . For large samples, therefore, Z can be chosen as an estimator of µ but since (Proposition 5.1b) Var(M) < Var(Z), the sample mean is more efﬁcient than Z. We open here a short parenthesis: the fact that the sample median is less efﬁcient than M should not lead the analyst to discard Z altogether as an estimator of µ. In fact, this statistic is much more robust than M and this quality is highly desirable in practice when the data may be contaminated by ‘outliers’. We do not enter in any detail here but we only say that ‘robust’ in this context means that Z, as an estimator of the mean, is much less sensitive than M to the presence of outliers, where the term ‘outlier’ denotes an unexpectedly high or low value which, at ﬁrst sight, does not seem to belong to the sample. As a matter of fact, this is often the case because outliers are generally due to recording, transmission or copying errors; in some cases, however, they may be true data of exceptional events. The interested reader can refer, for example, to Chapter 16 of [27]. Example 5.4(c) Turning brieﬂy to asymptotic properties it is immediate to show, for instance, that M and Ak are consistent estimators of µ and αk , respectively. In fact, they are unbiased and their variance – see eq. (5.10) and the second of (5.15) – satisfy condition (b) of Proposition 5.6. Also, having

Preliminary ideas and notions

185

σ2

S2

already noted that is an asymptotically unbiased estimator of we can use eq. (5.14) to determine that – if µ4 exists – then Var(S2 ) → 0 as n → ∞ and therefore S2 is a consistent estimator of σ 2 by virtue of Proposition 5.6. Example 5.4(d) As a ﬁnal example, let us suppose that the parent r.v. X of the sample is distributed according to a Cauchy pdf of the form f (x; θ ) =

1 π[1 + (x − θ )2 ]

(5.33)

which represents our parametric model. Suppose further that we consider the sample mean M as an estimator of θ. Now, using characteristic functions it is not difﬁcult to show that M has the same distribution as X and therefore the probability P(|M − θ| ≥ ε) – being the same for all n – cannot tend to zero as n → ∞. The conclusion is that M is not a consistent estimator of θ. 5.4.1

Cramer–Rao inequality

In the preceding section we deﬁned the relative efﬁciency of estimators by restricting our attention to the class u(θ ) of UEs. Within this class, the requirement of minimum variance – see eq. (5.28) – is the property of interest. Suppose, however, that somehow (we will have to say more about this later) we can ﬁnd some unbiased estimators of a given parameter θ. Among these estimators, we can select the most efﬁcient, but how do we know that there are no more efﬁcient ones? In many cases, the Cramer–Rao inequality can answer this question. In fact, provided that some ‘regularity conditions’ are satisﬁed, it turns out that the variance of UEs is bounded from below; if we ﬁnd an estimator whose variance equals this lower bound then we also know that this estimator – in the terms speciﬁed by Proposition 5.4 – is unique. In order to keep things relatively simple, we consider the one-dimensional continuous case and denote by f (x; θ ) the pdf of the parent r.v. X of the sample (X1 , . . . , Xn ). Then the so-called likelihood function L(x; θ ) = L(x1 , . . . , xn ; θ ) =

n

f (xi ; θ )

(5.34)

i=1

is the pdf of the sample. We assume the following regularity conditions: (a) the set {x : f (x; θ ) > 0} – that is, in mathematical terminology, the support of the pdfs f (x; θ ) – does not depend on θ; (b) thefunction f (x; θ) is differentiable with respect to θ; ∂ ∂ (c) ∂θ f (x; θ ) dx f (x; θ ) dx = ∂θ ∂ ∂ (d) ∂θ T(x)L(x; θ ) dx = T(x) ∂θ L(x; θ ) dx where all (Lebesgue) integrals are on all space (R in (c) and Rn in (d))

186 Mathematical statistics (e) E[U 2 (X; θ )] < ∞ where the ‘score’ or ‘contribution’ function U(X; θ ) is deﬁned as ∂ ∂ ln L(X; θ ) = ln f (Xi ; θ ) ∂θ ∂θ n

U(X; θ ) =

(5.35)

i=1

and the second equality descends from (5.34). Proposition 5.8 (Cramer–Rao inequality) ditions, let T = T(X) ∈ u(θ ). Then Var(T) ≥

1 E[U 2 (X; θ )]

=

Under the above regularity con-

1 In (θ )

(5.36)

where the function E[U 2 (X; θ )], being important in its own right, is denoted by In (θ) and called Fisher’s information (on θ) contained in the sample X. As a preliminary result, note that E[U(X; θ )] = 0

(5.37)

In fact, since f (X; θ ) = f (X1 ; θ ) = · · · = f (Xn ; θ ), from (5.35) it follows

∂ ∂ E[U(X; θ )] = ln f (Xi ; θ ) = nE ln f (X; θ ) E ∂θ ∂θ i ∂ ∂ = n f (x; θ ) [ln f (x; θ )] dx = n f (x; θ ) dx ∂θ ∂θ ∂ =n f (x; θ ) dx = 0 ∂θ where we used the relation ∂ ln f /∂θ = (1/f )(∂f /∂θ ) in thefourth equality, condition (c) in the ﬁfth and the last equality holds because f (x; θ ) dx = 1. Now, in order to prove eq. (5.36) we apply Cauchy–Schwarz inequality (eq. (3.21)) to the variables (T(X) − θ ) and U(X; θ ) E2 [(T − θ )U] ≤ E[(T − θ )2 ]E(U 2 ) = Var(T)E(U 2 )

(5.38)

Since E[(T − θ )U] = E(TU) − θE(U) = E(TU), we use the relation ∂ ln L/∂θ = (1/L)(∂L/∂θ ) and condition (d) to get ∂ ∂L(x; θ ) E(TU) = T(x)L(x; θ ) [ln L(x; θ )] dx = T(x) dx ∂θ ∂θ ∂ ∂ ∂ = E(T) = θ =1 T(x)L(x; θ ) dx = ∂θ ∂θ ∂θ

Preliminary ideas and notions

187

so that the l.h.s of (5.38) is unity and Cramer–Rao inequality follows. Furthermore, by considering the explicit form of U of eq. (5.35) we get ⎧ 2 ⎫

2 6 ⎬ ⎨ ∂ ∂ 2 In (θ)= E(U )= E = ln f (Xi ; θ ) ln f (Xi ; θ ) E ⎭ ⎩ ∂θ ∂θ i i

( ' ∂ ∂ + ln f (Xi ; θ ) ln f (Xj ; θ ) E ∂θ ∂θ i =j

2 6 ∂ = nI(θ ) = nE ln f (X; θ ) ∂θ where (i) the sum on i = j is zero because of independence and of eq. (5.37) and (ii) I(θ) = E{(∂ ln f (X; θ )/∂θ )2 } is called Fisher’s information and is the amount of information contained in one observation; the fact that In (θ) = nI(θ) means that the information of the sample is proportional to the sample size. In the light of these considerations we can rewrite (5.36) as Var(T) ≥

1 nI(θ )

(5.39)

If, in addition, f is twice θ-differentiable and we can interchange the signs of integration and derivative twice we have yet another form of the inequality. In fact, while proving eq. (5.37) we showed that (∂f /∂θ ) dx = 0; under the additional assumptions, we can differentiate with respect to θ to get 0=

∂ 2f dx = ∂θ 2

1 f

∂ 2f ∂θ 2

1 ∂ 2f f dx = E f ∂θ 2

and since we can use this last result to obtain

∂ 2 ln f ∂ 1 ∂f 1 ∂f 2 E = E = −E ∂θ f ∂θ ∂θ 2 f 2 ∂θ

∂ ln f 2 1 ∂ 2f +E = −E = −I(θ ) f ∂θ 2 ∂θ Cramer–Rao inequality can be written as "

∂ 2 f (X; θ ) Var(T) ≥ − nE ∂θ 2

#−1 (5.40)

188 Mathematical statistics A few remarks are in order: (1) the equal sign in (5.36) holds if and only if the two r.v. in Cauchy–Schwarz inequality are linearly related, that is, when T(X) − θ = a(θ )U(X; θ )

(5.41)

where a(θ) is some function of θ. (2) if T is an UE of a (differentiable) function τ (θ) of θ, the numerator of Cramer–Rao inequality becomes {τ (θ )}2 and eq. (5.41) becomes T − τ (θ ) = a(θ )U. Whenever this relation holds, then Var(T) = a(θ )E(TU). Then, since the θ-derivative of τ (θ) = E(T) = T(x)L(x; q) dx is τ (θ ) = E(TU), it follows that Var(T) = a(θ )τ (θ )

(5.42)

which reduces to Var(T) = a(θ ) if, as we considered above, τ (θ) = θ. (3) Cramer–Rao inequality establishes a lower bound for the variance of an UE; this does not imply that an estimator with such minimum variance exists (when this is not the case, one may use Bhattacharya’s inequality; for more details see for instance Ref. [17] or [19]). (4) the ratio between the lower bound and Var(T) is called efﬁciency of the estimator and denoted by eT , that is, eT =

1 1 = In (θ)Var(T) nI(θ )Var(T)

(5.43)

where 0 ≤ eT ≤ 1 and eT = 1 indicates a MVUE estimator. (5) The discrete case, with only minor modiﬁcations is analogous to the continuous one. Example 5.5(a) Consider a sample from a normal distribution with unknown mean θ = µ and known variance σ 2 . All regularity conditions are met and f (x; θ ) is given by eq. (2.29a). Then ∂ ln f (x; µ)/∂µ = (x − µ)/σ 2 and Fisher’s information is * I(µ) = E

∂ ln f (x; µ) ∂µ

+2 =

1 1 E[(X − µ)2 ] = 2 σ4 σ

which, as expected, implies that a smaller variance corresponds to a higher information. Now, considering M as an estimator of µ and knowing that (eq. (5.10)) Var(M) = σ 2 /n we get eM = 1; therefore M is a MVUE estimator of µ. Also, eq. (5.41) must hold. In fact, using the expression of f (Xi ; θ )

Preliminary ideas and notions

189

pertinent to our case, we get from eq. (5.35)

U(X; µ) =

n n 1 (Xi − µ) = 2 (M − µ) σ2 σ i=1

which, in fact, is eq. (5.41) where T − θ = M − µ and a(θ ) = σ 2 /n. Note that, in agreement with eq. (5.42), a(θ ) = Var(M). Example 5.5(b) Turning to a discrete case, consider a sample from a parent Poisson r.v. X with unknown parameter θ = λ. From the pmf of eq. (4.1) we get the Fisher’s information I(λ) = λ−1 and since Var(X) = λ implies Var(M) = λ/n, we have again eM = 1. (Incidentally, it is not out of place to point out that examples (a) and (b) must not lead to the (wrong) conclusion that M – although always unbiased – is always an efﬁcient estimator of the mean.) Example 5.5(c) Exponential Models. An important class of parametric models has the general form f (x; θ ) = exp{A(θ )B(x) + C(θ ) + D(x)}

(5.44)

and is called exponential. Not all exponential models satisfy the regularity conditions, but for the ones that do the following considerations apply. Denoting by a prime the derivative with respect to θ, the score function is easily obtained as U(X; θ ) = A (θ )

n i=1

"

C (θ ) 1 B(Xi ) + nC (θ ) = nA (θ ) B(Xi ) + n A (θ ) n

#

i=1

which corresponds to eq. (5.41) once we set (see also remark (2)) T(X) = n−1

n

B(Xi )

i=1

τ (θ) = −C (θ )/A (θ )

(5.45)

a(θ) = [nA (θ )]−1 from which it follows that for the exponential class the statistic T(X) is an efﬁcient estimator of τ (θ ), where T(X) and τ (θ ) are given by the ﬁrst and

190 Mathematical statistics second parts of eq. (5.45). This, by eq. (5.42), implies Var(T) =

τ (θ ) nA (θ )

(5.46)

Also, only a small effort is required to show that I(θ) = τ (θ )A (θ )

(5.47)

By appropriately identifying the functions A, B, C and D, many practical models are, as a matter of fact, exponential. Examples (a) and (b), for instance, are two such cases. In fact, if we set A(θ ) = θ/σ 2 , B(x) = x, C(θ) = −θ 2 /2σ 2 and D(x) = −x2 /2σ 2 we ﬁnd example (a) and, as above, we determine that M is an efﬁcient estimator of µ with Var(M) = σ 2 /n. The Poisson example of case (b), on the other hand, is obtained by setting A(θ) = ln θ, B(x) = x, C(θ ) = −θ and D(x) = − ln x!. Example 5.5(d) As a further special case of exponential model, the reader is invited to consider a sample from a normal population with known mean 2 = σ 2 . By setting A(θ ) = −1/2θ 2 , B(x) = µ and unknown variance θ√ 2 (x − µ) and C(θ ) = − ln(θ 2π ) it turns out that T(X) = n−1 (Xi − µ)2 is an efﬁcient estimator of τ (θ ) = θ 2 . Also, the reader should check that 2 −1 3 −1 2 eq. (5.41) for this case is n i (Xi − µ) − θ = n θ U(X; θ ) and that 4 Var(T) = 2θ /n, in agreement with result (ii) of Example 5.4(a). The above examples show that for large samples the order of the variance of UEs is n−1 . This, as a matter of fact, is a general rule which applies to regular models. It is worth pointing out that in some cases of non-regular models it is possible to ﬁnd UEs whose variance decreases more quickly than n−1 as n increases – that is, we can ﬁnd UEs with variances smaller than the Cramer–Rao limit. Examples of these ‘superefﬁcient’ estimators can be found, for instance, in Chapter 32 of [3] or in Chapter 2 of [19]. In closing this section, we brieﬂy outline the case of more than one parameter, let us say k, so that q = (θ1 , θ2 , . . . , θk )T is a k-dimensional vector. Then, the score function is itself a vector U = (U1 , . . . , Uk )T where Ui (X; q) = ∂ ln L(X; q)/∂θi and one can form the k × k information matrix of the sample as In (q) = E(UUT ) = nI(q)

(5.48)

(the second equality is the vector counterpart of the one-dimensional relation In (θ) = nI(θ), valid in our experiment of repeated independent trials). The ijth element Iij (q) of I(q) – which, in turn, is the information matrix of one

Preliminary ideas and notions

191

observation – is given by (i, j = 1, 2, . . . , k)

∂ ln f (X; q) ∂ ln f (X; q) Iij (q) = E ∂θi ∂θj

∂ 2 ln f (X; q) = −E ∂θi ∂θj

(5.49)

and the last relation holds if f (x; q) is twice differentiable with respect to the parameters θ1 , . . . , θk . Clearly, both In and I are symmetric, that is, In = InT and I = IT . Given these preliminary notions, let T(X) be an unbiased estimator of some function τ (q) = τ (θ1 , . . . , θk ) of the unknown parameters; then the Cramer–Rao inequality is now written Var(T) ≥ dT In−1 d =

1 T −1 d I d n

(5.50)

where d = d(q) is the vector of derivatives d(q) = (∂τ/∂θ1 , . . . , ∂τ/∂θk )T and, similarly to eq. (5.41), the equality holds if and only if T(X) − τ (q) = [a(q)]T U(X; q)

(5.51)

for some vector function a = (a1 , . . . , ak )T of the parameters (note that, in general, ai = ai (q) for all i = 1, . . . , k). Moreover, as in the one-dimensional case, one calls efﬁcient an estimator of τ (q) whose variance coincides with the r.h.s. of (5.50). Finally, since it is evident that eq. (5.50) holds only if In (q) (and therefore I(q)) is non-singular for all q ∈ , this assumption is generally added to the other deﬁning conditions of regularity. 5.4.2

Sufﬁciency and completeness of estimators

In order to evaluate the ‘goodness’ of an estimator, another desirable property – besides the ones considered so far – is sufﬁciency. The deﬁnition is: given an unknown parameter θ, an estimator T(X) of θ is sufﬁcient (or exhaustive for some authors) if the conditional likelihood L(x|T = t; θ ) does not depend on θ. Equivalently, T is sufﬁcient if the conditional probability Pθ (X ∈ A|T = t) does not depend on θ for any event A ⊂ . This deﬁnition is not self-evident and some further comments may help. In essence, sufﬁciency requires that the values t = T(x1 , . . . , xn ) taken on by the statistic T must contain all the information we can get on θ. In other words, suppose that two realizations of the sample x and x both lead to the value t = T(x) = T(x ). If the function L(x|T = t; θ ) depended on θ then we would have, say, L(x|T = t) > L(x |T = t) for θ ∈ 1 and L(x|T = t) < L(x |T = t) for θ ∈ 2 , where 1 ∪ 2 = and 1 ∩ 2 = ∅ (i.e. the sets 1 , 2 form a partition of the parameter space ). Therefore, knowing which one of the two realization has occurred provides more information than just the fact of knowing that T = t. So, for instance, if x has occurred,

192 Mathematical statistics we would tend to think that, preferably, θ ∈ 1 . If, on the other hand L(x|T = t) = L(x |T = t) for all θ ∈ then the speciﬁc realization of the sample leading to T = t is irrelevant and – for a ﬁxed sample size n – the equality T = t summarizes all that we can know in order to estimate θ. This is why T is a ‘sufﬁcient’ estimator of θ. In practice, it may be difﬁcult to determine sufﬁciency just by using the deﬁnition above. Often, an easier way to do it is to use Neyman’s theorem (which some authors give as the deﬁnition of sufﬁciency) Proposition 5.9(a) (Neyman’s factorization theorem) A statistic T(X) is sufﬁcient for θ if and only if the likelihood function can be factorized into the product of two functions g(T(x); θ ) and h(x), that is, L(x; θ ) = g(T(x); θ )h(x)

(5.52)

(where it should be noted that the factor g depends on x only through T(x)). In fact, since L(x|t; θ ) =

Pθ (X = x ∩ T = t) L(x; θ ) = L(x ; θ ) Pθ (T = t)

(the sum at the denominator is over all realizations x giving T = t) if we assume that the factorization (5.52) holds we get L(x|t; θ ) = h(x)/ h(x ) and therefore, according to the deﬁnition above, T is sufﬁcient (if x is such that T(x) = t then L(x|t; θ ) = 0; consequently L(x|t; θ ) does not depend on θ for any realization of the sample). The proof of the reverse statement – that is, if L(x|t; θ ) does not depend on θ then eq. (5.52) holds – is left to the reader. Example 5.6(a) Let X be a sample from a Poisson variable (see eq. (4.1)) of unknown parameter θ. Then L(x; θ ) =

n

i=1

e−θ

θ xi θ x1 +···+xn = e−nθ xi ! x1 ! · · · xn !

(5.53)

and eq. (5.52) holds with g = e−nθ θ x1 +···+xn and h = (x1 ! · · · xn !)−1 . It follows that the statistic T(X) = X1 + · · · + Xn is a sufﬁcient estimator of θ. Alternatively, in this case we could also use the deﬁnition by noting (Section 4.2) that T is itself a Poisson variable of parameter nθ. So, Pθ (T = t) = {e−nθ (nθ )t }/t! and L(x|t; θ ) is independent on θ because L(x|t; θ ) =

e−nθ θ t t! = t {Pθ (T = t)}x1 ! · · · xn ! n (x1 ! · · · xn !)

Preliminary ideas and notions

193

Example 5.6(b) Let X be a sample from a Gaussian variable with unknown mean µ = θ and known variance σ 2 . Then, deﬁning T(x) = x1 + · · · + xn we have n

1 (xi − θ )2 L(x; θ ) = exp − √ 2σ 2 2π σ i=1 (5.54)

n 1 1 = √ x2i − 2θT(x) + nθ 2 exp − 2 2σ 2π σ i

and since Neyman’s theorem holds by choosing +

*

1 g(T(x); θ ) = exp − 2 (2θT(x) + nθ 2 ) 2σ √ 1 2 −n h(x) = ( 2π σ ) exp − 2 xi 2σ i

the statistic T(X) = X1 + · · · + Xn is a sufﬁcient estimator of θ. A corollary to Proposition 5.9(a) is Proposition 5.10 (i) If the function z is one-to-one and T is sufﬁcient for θ, then Z = z(T) is also a sufﬁcient estimator of θ. Moreover, (ii) Z = z(T) is a sufﬁcient estimator of θˆ = z(θ ). In fact, the relation L(x; θ ) = g(z−1 (Z); θ )h(x) = g1 (Z; θ )h(x) proves part (i) while part (ii) follows easily by also considering the relation θ = z−1 (θˆ ). An immediate consequence of the corollary is that if T(X) = X1 + · · · + Xn is a sufﬁcient estimator for the mean µ of a population, so is M = T/n. At this point, an important observation is that Neyman’s factorization (5.52) implies eq. (5.41) which – as we have determined – characterizes efﬁcient (MVUE) estimators. In other words, this means that the class of sufﬁcient statistics (for the parameter θ) includes the MVUE of θ when this estimator exists (note, however, that sufﬁcient statistics may exist even when there is no MVUE). Moreover, Rao–Blackwell theorem states that the following: Proposition 5.11 (Rao–Blackwell) of a sufﬁcient statistic.

The MVUE, when it exists, is a function

In fact, let X be a sample from a population with an unknown parameter θ, T(X) a sufﬁcient statistic for θ and T1 (X) an arbitrary UE of θ. Then E(T1 |T) (note that in strict symbolism we should write Eθ (T1 |T)) is a function of the form H(T) which takes on the value H(t) = E(T1 |t) when T = t. Since

194 Mathematical statistics T, T1 are random variables in their own right, we can use eq. (3.89a) to get E[H(T)] = E[E(T1 |T)] = E(T1 ) = θ where the last equality holds because T1 is unbiased. The consequence is that H(T) is itself an UE of θ. In addition to this, eq. (3.91) shows that Var(T1 ) = E[Var(T1 |T)] + Var[H(T)] which – since E[Var(T1 |T)] ≥ 0 – implies Var[H(T)] ≤ Var(T1 )

(5.55)

(the equal sign holds if and only if E[Var(T1 |T)] = E{[T1 − E(T1 |T)]2 } = 0, that is, whenever T1 = H(T) – or, more precisely, when P{T1 = H(T)} = 1). At this point one could conclude that H(T) is (i) an UE of θ and (ii) more efﬁcient than T1 . Before doing this, however, one must show that H(T) is a statistic, that is, does not depend on θ. By recalling eq. (3.88) we can write H(t) = E(T1 |t) =

T1 (x)L(x|t; θ ) dx

and note that both L(x|t; θ ) and T1 (x) do not depend on θ because, respectively, T is sufﬁcient and T1 is a statistic. So, H(t) does not depend on θ; moreover, as t varies the r.v. H(T) takes on the values H(t) with a density fT (t) which is itself independent on θ (T is a statistic). Consequently, as desired, H(T) is a statistic. Despite its intrinsic importance, Rao–Blackwell theorem is of little help in explicitly ﬁnding the MVUE (assuming that it exists). In fact, given a sufﬁcient and an unbiased estimator, T and T1 respectively, we can construct the UE H(T) which – although more efﬁcient than T1 – may not be the MVUE of θ. In principle, by using H(T) and another sufﬁcient statistic, we expect to be able to ﬁnd an even more efﬁcient (than H(T)) UE. However, if the original sufﬁcient statistic is complete (see deﬁnition below), it turns out that H(T) is the MVUE of the parameter θ. This is stated in the following proposition: Proposition 5.12 (Lehmann–Scheffé theorem) Let T(X) be a sufﬁcient and complete statistic for θ and T1 (X) an UE of θ. Then H(T) = E(T1 |T) is the efﬁcient estimator of θ. Before showing why this is so, we give the deﬁnition of completeness: a sufﬁcient statistic T is complete if for any (bounded) function ϕ(T) the relation Eθ [ϕ(T)] = 0

for all θ ∈

implies ϕ(t) = 0 for almost all values t = T(x) (the term ‘almost all’ refers to the measure Pθ and indicates that Pθ {ϕ(T(x)) = 0} = 1 for all θ ∈ ).

Preliminary ideas and notions

195

Returning to Proposition 5.12, assume that there exists another UE K(T) depending on T. Deﬁning L(T) = H(T) − K(T), we have E[L(T)] = θ − θ = 0 for all θ and this, by completeness, implies H(t) = K(t) a.e. which, in turn, shows that H(T) is the unique UE depending on T. Let now T˜ be ˜ an arbitrary UE. By virtue of the considerations above J(T) = E(T|T) is ˜ ˜ unbiased, Var[J(T)] ≤ Var(T) and the equality holds iff T = J(T). Since H(T) is the only UE depending on T, then we must have J(T) = H(T) and this proves the theorem. At this point, two closing remarks on sufﬁciency are worthy of mention. First we outline the generalization to the case of k unknown parameters. In this case the following deﬁnition applies: the vector T = (T1 , . . . , Tk ) is called a (jointly) sufﬁcient statistic for q = (θ1 , . . . , θk ) if the function L(x|t1 , . . . , tk ; q) does not depend on q. Neyman’s theorem, on the other hand, becomes: Proposition 5.9(b) The k-dimensional statistic T(X) = (T1 (X), . . . , Tk (X)) is (jointly) sufﬁcient for q = (θ1 , . . . , θk ) if and only if the likelihood function can be expressed as the product L(x; q) = g(T1 (x), . . . , Tk (x); q)h(x)

(5.56)

So, for example, it is easy to show that T = (T1 , T2 ) – where T1 = i Xi 2 and T2 = i Xi – is a sufﬁcient statistic for the two-dimensional Gaussian model with unknown mean and variance. Using the sufﬁcient statistic T we can then construct the well-known estimators M = n−1 T1 and S¯ 2 = (n − 1)−1 [T2 − n−1 T12 ] (see eq. (5.31)) of µ and σ 2 . The second and ﬁnal remark may appear rather obvious at ﬁrst glance but – we believe – deserves to be stated explicitly: sufﬁciency depends on the adopted statistical model. In other words, if the model is changed, a given sufﬁcient statistic may no longer be sufﬁcient in the new model. As a consequence, we should never discard the raw data and replace them with sufﬁcient statistics. In fact, although the main advantage of sufﬁcient statistics is to reduce the dimensionality of the sample without losing any information on the unknown parameter(s), it should also be kept in mind that the sample itself X = (X1 , . . . , Xn ) is always a sufﬁcient statistic irrespective of the adopted statistical model. Consequently – since the model may always be changed in the light of new evidence or of new assumptions – it is always good practice to preserve the original data. As an example, consider a sequence of binomial trials with unknown probability of success θ = p. The order of successes and failures is clearly unimportant in a model of independent trials and the sufﬁcient statistic T = X1 + · · · + Xn is equivalent to the sample as far as the estimation of θ is concerned. However, it can be shown [8] that it is not so if a new model of dependent trials is postulated.

196 Mathematical statistics (Incidentally, under the assumption of binomial independent – that is, Bernoulli – trials, the reader is invited to show that T = X1 + · · · + Xn is, indeed, a sufﬁcient statistic.)

5.5

Maximum likelihood estimates and some remarks on other estimation methods

In regard to point estimation, not much has been said so far on the way in which we can ﬁnd ‘good’ estimators although, in the preceding section, we have implicitly given a method of ﬁnding the MVUE of a parameter θ by using an UE T1 , a sufﬁcient complete statistic T and calculating the conditional expectation E(T1 |T). This procedure, however, often involves computational difﬁculties and is seldom used in practice. Other methods, in fact, have been devised and the most popular by far is the so-called ‘method of maximum likelihood’, introduced by Fisher in 1912 (although the deﬁnition of likelihood, also due to Fisher, appeared later). Before considering this, however, it is worth spending a few words on other methods with the main intention of simply illustrating – without any claim of completeness – other approaches to the problem. One of the oldest estimation procedures is Pearson’s ‘methods of moments’ and consists in equating an appropriate number of sample moments to the corresponding population moments which, in turn, depend on the unknown parameters. By considering as many moments as there are parameters, say k, one solves the resulting equations for θ1 , . . . , θk thus obtaining the desired estimates. In mathematical terms, if j = 1, . . . , k and aj = Aj (x) are the sample moments of the observed realization x = (x1 , . . . , xn ), one must solve the set of equations αj (θ1 , . . . , θk ) = aj ,

j = 1, 2, . . . , k

(5.57a)

whose result is in the form θj = tj (a1 , . . . , ak ),

j = 1, 2, . . . , k

(5.57b)

where the tj ’s – that is, the values taken on by the estimators Tj ’s at X = x – are obtained as functions of the sample moments. Recalling the developments of Section 5.3.1, this last observation on the Tj implies, under fairly general conditions, two desirable properties: for large samples the Tj are (i) consistent and (ii) asymptotically normal. Often, however, they are biased and their efﬁciency, as Fisher himself has pointed out [9] may be rather poor. For small samples, moreover, it should be kept in mind that sample moments may signiﬁcantly differ from their population counterparts, thus leading to poor estimates. This is especially true if higher-order moments must be used because in these cases n < 100 is generally considered a small sample.

Preliminary ideas and notions

197

As an example of the method, consider a sample X from a population with unknown mean µ = α1 and variance σ 2 = α2 − α12 . Equations (5.57a) and (5.57b) are simply αj = aj (j = 1, 2), and since a1 = m = n−1 i xi and a2 = n−1 i x2i , we get t1 = m and t2 = a2 − m2 = n−1 i (xi − m)2 . The desired estimators are therefore T1 = M T 2 = A2 − M 2 =

1 (Xi − M)2 n i

where we already know (eq. (5.12)) that T2 = S2 is a biased estimator of σ 2 . In all, however, the method has the advantage of simplicity and the ‘moments-estimates’ can be used as a ﬁrst approximation in view of a more reﬁned analysis. A second method of estimation is based on Bayes’ formula (eq. (3.79) in the continuous case). If we consider the unknown parameter θ as a value taken on by a r.v. Q with pdf fQ (θ ) – which, somehow, must be known by some prior information and for this reason is called ‘a priori’ density – Bayes’ formula yields (taking eq. (3.80b) into account) f (θ|x) = ∞

f (x|θ )fQ (θ )

−∞ f (x|θ )fQ (θ ) dθ

(5.58)

Then, by deﬁning Bayes’ estimator (of θ) as TB ≡ E(Q|X), its value tB corresponding to the realization x is taken as the estimate of θ, that is, ∞ tB = E(Q|X = x) =

θf (θ|x) dθ

(5.59)

−∞

A few additional comments on this method are worthy of mention. First, the function f (x|θ ) at the numerator of (5.58) is just the pdf f (x; θ ) that speciﬁes the statistical model. However, the point of view is different; instead of seeing θ as a deterministic quantity and postulating the existence of a ‘true’ value θ0 which – were it known – would provide the ‘exact’ probabilistic description by means of f (x; θ0 ), the Bayesian approach considers θ as a random variable and writes f (x|θ ) to mean that the realization x is conditioned by the event Q = θ. In this light, the ‘a posteriori’ density f (θ|x) provides information on θ after the realization x has been obtained and consequently we can use it to calculate the quantity E(Q|X = x) which, in turn – being the mean value of Q given that the event X = x has occurred – is a good candidate as an estimate of θ. Nonetheless, a key point of the method is how well we know fQ (θ). This, clearly, depends on the speciﬁc case under study although

198 Mathematical statistics it has been argued that a uniform distribution for Q may be used in cases of no or very little prior information (a form of the so-called ‘principle of indifference’). We do not enter into the details of this debated issue, which is outside the scope of the book, and pass to the main subject of this section: the method of maximum likelihood. Consider the statistical model (5.1) with k unknown parameters q = (θ1 , . . . , θk ). Once a realization of the sample x has been obtained, the likelihood L(x; q) is a function of q only; consequently, we can write L(q) and note that this function expresses the probability (density) of obtaining the result that, in fact, has been obtained, that is, x. In this light it is reasonable to assume as ‘good’ estimates of the unknown parameters the values qˆ = (θˆ1 , . . . , θˆk ) that maximize L(q), that is, ˆ = Max L(q) L(q)

(5.60)

q∈

where it should be noted that the maximum is taken on the parameter space and not on all the possible values that make mathematical sense for L(q). Owing to (5.60), θˆ1 , . . . , θˆk are called maximum likelihood (ML) estimates of θ1 , . . . , θk . As x varies, we will obtain different values of qˆ and this correspondence leads to the deﬁnition of ‘maximum likelihood estimators’ (MLE) as those statistics Tˆ 1 (X), . . . , Tˆ k (X) which, respectively, take on the values θˆ1 , . . . , θˆk when X = x. In practice, the ML estimates are obtained by ﬁnding the maximum of the log-likelihood function l(q) = ln L(q) (which is equivalent to maximizing L(q)), that is, by ﬁrst solving the likelihood equations ∂l(θ1 , . . . , θk ) = 0, ∂θj

j = 1, 2, . . . , k

(5.61)

and then checking which solution is an absolute maximum (in fact, the solutions of eq. (5.61) – if there are any – determine the stationary points of l(q), which can be minima, maxima or saddle points). The whole procedure is generally rather easy if we have one (or two) unknown parameter(s) but it is evident that computational difﬁculties may arise for higher values of k. In these cases one must resort to numerical techniques of solution of eq. (5.61) and the Newton–Raphson iteration method is frequently used for this task. The subject, however, is beyond our scope and the reader interested in computational aspects may refer, for instance, to [29] (Incidentally, in regard to the determination of the maximum among the solutions of (5.61), it may be worth recalling a theorem of analysis which states the following: If l(q) is twice differentiable and is an open set of Rk , a maximum is attained ˆ T H(q)(q ˆ ˆ deﬁned by the Hessian matrix at qˆ if the quadratic form (q − q) − q) 2 H(q) = [∂ l/∂θi ∂θj ] (i, j = 1, . . . , k) is negative deﬁnite).

Preliminary ideas and notions

199

Example 5.7(a) Considering a sequence of n Bernoulli trials, the statistical model is clearly given by (5.2). Then, the ML estimate of the parameter θ = p is easily obtained by ignoring the terms with the factorials (which do not involve θ) and writing l(θ ) = x ln θ + (n − x) ln(1 − θ ). Taking the derivative ∂l x n−x = − =0 ∂θ θ 1−θ we get the solution θˆ = x/n, which is a maximum because ∂ 2 l/∂θ 2 is negˆ Also note that the ML estimate coincides with the ative at the point θ = θ. observed frequency of success. This speciﬁc example is one among many others that, a posteriori, justiﬁes the relative frequency approach to probability discussed in Chapter 1. Example 5.7(b) In the case of a sample from a normal population with unknown mean µ = θ1 and variance σ 2 = θ2 , the reader is invited to ˆ1 = n−1 i xi = m and θˆ2 = determine that the ML estimates are θ n−1 i (xi − m)2 = s2 so that the MLE are M and S2 , respectively. The examples above do not do justice to the ML method because the reader can easily check that the method of moments yields the same estimators. In general, however, this is not so and the reason why the ML method is so widely adopted lies in the good properties of MLEs. The ﬁrst can be called the ‘covariance’ property with respect to parameter transformations; in fact, referring to eq. (5.4) we have Proposition 5.13 If qˆ = (θˆ1 , . . . , θˆk ) is the MLE of q = (θ1 , . . . , θk ) and h a ˆ is the MLE of h(q). one-to-one mapping from to (, ⊂ Rk ), rˆ = h(q) The proof is immediate because the function h−1 : → exists and Max L(q) = Max L(h−1 (r)) ≡ Max Lr (r) q∈

r∈

r∈

(we note in passing that the explicit form of Lr is obtained by simply setting h−1 (r) in the original likelihood function L; the differential elements must not be included because we transform the parameters and not the variables). So, for instance, the fact that S2 is the MLE of σ 2 in a,normalmodel with known mean and unknown variance tells us that S = {n−1 (Xi − µ)2 } is the MLE of the standard deviation σ . A useful consequence of Proposition 5.13, moreover, is that some problems can be cast in a simpler form by an appropriate change of parameters; in these cases we can solve the simpler problem – thus ﬁnding the ML estimates rˆ = (ˆr1 , . . . , rˆk ) – and then determine q = (θ1 , . . . , θk ) by means of h−1 . A nice example of this is given in Ref. [19]

200 Mathematical statistics (Chapter 2, Example 2.22) where a bivariate normal model (see eqs (3.61a) and (3.61b)) is considered and the ML estimates of σ 2 = θ1 and ρ = θ2 are determined by introducing the new parameters r1 = −[2σ 2 (1 − ρ 2 )]−1 and r2 = ρ[σ 2 (1 − ρ 2 )]−1 . Then, the desired result σˆ 2 = (2n)−1 ρˆ = 2

x2i + yi2

i

xi y i /

i

x2i + yi2

(5.62)

i

is obtained with a noteworthy simpliﬁcation of the calculations. A ﬁnal remark on Proposition 5.13: some authors speak of ‘invariance’ property. This term, however, would imply that the MLEs remain unchanged; since, in fact, they do change according to the transformation law h, we think that the term ‘covariance’ should be preferred. Other properties concern the relation between MLEs, efﬁcient estimators and sufﬁcient statistics, stated by the following two results, respectively. Proposition 5.14

ˆ If a MVUE T(X) of θ exists, then T(X) = T(X).

For regular problems, in fact, if a MVUE of θ exists it satisﬁes eq. (5.41). This, together with the likelihood equation (5.61) yields the desired result. ˆ Proposition 5.15 If T(X) is a sufﬁcient statistic for θ and the MLE T(X) ˆ of θ exists and is unique, then T is a function of T. The proof is almost immediate: since T is sufﬁcient, Neyman’s factorization (5.52) holds and maximizing L is equivalent to maximizing g which, in turn, depends on T. Consequently, the MLE itself will be a function of T. Before turning to the asymptotic properties of MLEs – which will be the subject of the next section – we point out two facts and state without proof an interesting result worthy of mention. First, MLEs, although asymptotically unbiased (see the following section), are often biased. Second, the ML method can be used in cases more general than the one considered here, that is, independent drawings from a ﬁxed distribution. For instance, the example (taken from Ref. [19]) on parameter transformation and mentioned above is a case in which, in fact, independence does not hold. Finally, the following proposition [30] provides an interesting characterization of some probability distributions based on a ML estimate: Proposition 5.16 For n ≥ 3, let X be a sample from a continuous population with pdf of the form f (x − θ ) and let X(1) ≤ X(2) ≤ · · · ≤ X(n) be the corresponding order statistics. If Tˆ = i ai X(i) with ai ≥ 0 and a1 + · · · + an = 1 is the MLE of θ, then

Preliminary ideas and notions

201

(a) if a1 = · · · = an = 1/n, then f is a normal density; (b) if a1 + an = 1; a1 an > 0, then f is a uniform density; (c) if aj + aj+1 = 1; aj aj+1 > 0 with j ∈ {1, 2, . . . , n − 1}, then f is a Laplace density. In regard to point (c), we call a Laplace r.v. a continuous r.v. X whose pdf is

|x − α| 1 fX (x) = exp − 2β β

(5.63)

where x ∈ R and the two parameters are such that α ∈ R; β > 0. Its CF is ϕX (u) =

eiαu 1 + β 2 u2

(5.64)

while its mean and variance are, respectively E(X) = α Var(X) = 2β 2 5.5.1

(5.65)

Asymptotic properties of ML estimators

As a matter of fact, some important properties of MLEs are asymptotic in nature. Since their proofs, however, are generally rather lengthy, this section is limited to the statement of the main results. For details, the interested reader can refer to more specialized literature (see, for instance, [3, 17, 19, 26, 28]). Assuming, as it is often the case, that we are dealing with a regular problem (Section 5.4.1) and that the likelihood function Ln attains its maximum at an interior point of for all n (this, in other words, means that the MLE exists for all n), then: (1) Tˆ n → θ[P], that is, the MLE is (weakly) consistent; √ (2) the r.v. n(Tˆ n − θ ) converges in distribution to a normal r.v. with zero mean and variance 1/I(θ ) or, equivalently, the MLE Tˆ n is asymptotically normal with mean θ and variance given by the Cramer–Rao limit {nI(θ)}−1 (eq. (5.39)). Although we do not provide the proofs of the above statements, some comments are not out of place. First, it should be noted that result (1) can be strengthened and strong consistency (in the sense of a.s. convergence) can be proven (see, for instance, [1]). Second, we have noted in the preceding section that the ML method does not always lead to unbiased estimators;

202 Mathematical statistics they are, however, asymptotically unbiased because the bias – which, anyway, can generally be removed for ﬁnite values of n – tends to zero as n−1 when we let n → ∞. Besides the minor inconvenience of bias for ﬁnite n, a more important property is given in point (2) in regard to the variance of MLEs. In fact, if we introduce the notion of asymptotic efﬁciency e¯ T of an estimator T as eT = limn→∞ eT , then eTˆ = 1, meaning that MLEs are asymptotically efﬁcient. Now, this fact does not imply that MLEs are the only asymptotically normal and asymptotically efﬁcient estimators but it has been shown that, in general, MLEs have better efﬁciency properties for large values of n (Refs. [23, 24]). In regard to this last observation, we note in passing that (2) does not generally imply that Var(Tˆ n ) → {nI(θ )}−1 as n → ∞ (D-convergence does not imply convergence of the moments); however, for a large class of asymptotically normal estimators the variance can be expressed as Var(T) =

a2 (θ ) 1 + ··· + nI(θ ) n2

and the estimator with the minimum a2 (θ ) is to be preferred (second-order efﬁciency). Quite often it turns out that MLEs are such estimators. Another remark on result (2) is that cases where the asymptotic variance depends on the unknown parameter are rather common. An appropriate parameter transformation can ﬁx the problem by maintaining, at the same time, asymptotic normality. In fact, if h is a differentiable function with h = √ 0 then it can be shown that the variable n{h(Tˆ n ) − h(θ )} is asymptotically normal with zero mean and variance [h (θ )]2 /I(θ ). Enforcing the condition √ that this new variance equals a constant – say b2 – we get h (θ ) = b I(θ ) and therefore h(θ) = a + b

, I(θ ) dθ

(5.66)

where both constants a, b can be chosen so that h(θ ) is in simple form. Example 5.8(a) Consider a sample from a Poisson variable (eq. (4.1)) with unknown parameter λ = θ. Since ∂ 2 f /∂θ 2 = −x/θ 2 then I(θ) = E(x/θ 2 ) =

1 1 θ x e−θ = x 2 x! θ θ x

(5.67)

because the sum – being the mean of the parent r.v. X – equals θ. It follows from (5.67) that the Cramer–Rao limit is θ/n. On the other hand, the MLE of θ is obtained by taking the logarithm of eq. (5.53) and equating its derivative

Preliminary ideas and notions

203

to zero; the reader can easily check that the result is 1 Xi Tˆ n = M = n

(5.68)

i

whose variance is θ/n (eq. (5.10), taking into account that Var(X) = θ). So, as expected, the MLE is consistent and in this case it is also efﬁcient because (eq. (5.43)) eTˆ = 1. Moreover, from result (2) we know that √ ˆ n(Tn − θ ) is asymptotically normal with zero mean and a variance which depends on the parameter, that is, 1/I(θ ) = θ. Setting a = 0) and b = 1 in √ √ √ eq. (5.66) we get h(θ ) = 2 θ so that the new variable 2 n( Tˆ n − θ ) is asymptotically standard-normal, that is, with zero mean and unit ) variance. √ √ Alternatively, setting a = 0 and b = 1/2 we have that Yn = n( Tˆ n − θ ) is asymptotically normal with zero mean and Var(Yn ) = 1/4. Example 5.8(b) When the model is non-regular, asymptotic normality may not hold. As an example, in the uniform model f (x; θ ) = 1/θ for 0 ≤ x ≤ θ (and zero otherwise) the likelihood function is

L(x; θ ) =

⎧ ⎨1/θ n ,

x(n) ≡ max xi ≤ θ

⎩0,

otherwise

1≤i≤n

and T = X(n) – where X(n) is the nth order statistic – is a sufﬁcient statistic for θ. Also, the likelihood function is monotone decreasing for θ ≥ x(n) and therefore it attains its maximum at θ = x(n) where, however, there is a discontinuity. So, even if we can call T = X(n) the MLE of θ, this is not a solution of the likelihood equation (5.61) and we may not expect property (2) to hold. In fact, we already know from Section 5.3.1 that the extreme value of the sample X(n) is not asymptotically normal. The above results still hold in the case of several parameters. Explicitly, referring to the considerations at the end of Section 5.4.1, property (2) becomes √ ˆ (2 ) the r.v. n(T n − q) is asymptotically normal with zero mean and variance {I(q)}−1 or, in case we are estimating a scalar function τ (q) = τ (θ1 , . . . , θk ) of the unknown parameters: √ (2 ) the r.v. n{Tˆ n − τ (q)} is asymptotically normal with zero mean and variance dT I−1 d, where d(q) = (∂τ/∂θ1 , . . . , ∂τ/∂θk )T .

204 Mathematical statistics

5.6

Interval estimation

Within the framework of the statistical model (5.1), we have discussed in the preceding sections the subject of ‘point estimation’ which, in essence, consists in (a) ﬁnding a ‘good’ estimator T(X) of the unknown parameter θ and (b) using the data from the experiment – that is, the realization of the sample x – to calculate the numerical value t = T(x). Then, on the basis of a number of considerations on what is meant by ‘good’, we expect t to be a reliable estimate of θ (broadly speaking, we could call it our educated ‘best-guess’ on the true value of θ). The procedure above is well justiﬁed if the main question of the estimation problem is ‘what value should I use for θ?’. If, however, one is more interested in specifying a range of values within which he/she can conﬁdently expect θ to lie, then the method of ‘interval estimation’ provides a better way to tackle the problem. In perspective, moreover, one should consider that a point estimate is almost meaningless without a statement of its ‘reliability’. So, still keeping the model (5.1) as our starting point, we now wish to determine an interval which contains the true value of θ – though unknown – at a speciﬁed ‘conﬁdence level’ (CL for short) γ = 1−α (0 < γ < 1). This, in other words, means that we have to ﬁnd two statistics T1 , T2 , with T1 < T2 , such that Pθ {T1 (X) < θ < T2 (X)} = γ

(5.69a)

for all θ ∈ . In this case we call (T1 , T2 ) a γ -conﬁdence interval (often γ -CI) for θ and T1 , T2 , respectively, the lower and upper conﬁdence limits. Note that eq. (5.69a) deﬁnes a random interval which, on the one hand, depends on the sample X but, on the other hand, does not depend on θ (because both limits are statistics). By carrying out an experiment we obtain a realization of the sample x and, accordingly, the values t1 = T1 (x) and t2 = T2 (x) for the two statistics; the interval (t1 , t2 ) is then an estimate of the γ -CI. At this point, one could be tempted to say that θ belongs to (t1 , t2 ) with a probability γ . This statement, however, is wrong because (t1 , t2 ) is not a random interval and therefore the true value of θ either belongs to it or it does not. The correct interpretation must be given in terms of relative frequency of success: if the experiment is repeated many times – thus obtaining many estimates of (T1 , T2 ) – the resulting estimated intervals will contain the true value of θ in 100γ % of the cases. Conversely, in the long run we will be wrong in 100α% of the cases. This, in essence, is the meaning of the term ‘conﬁdence’ in this context. Now, before showing how to determine the conﬁdence limits, some additional remarks on eq. (5.69a) are in order: (i) If the population under study is discrete it may not be possible to meet condition (5.69a) exactly; in this case we call γ -CL the smallest interval

Preliminary ideas and notions

205

such that Pθ {T1 (X) < θ < T2 (X)} ≥ γ

(5.69b)

for all θ ∈ . (ii) The statistic Dγ (X) = T2 − T1 is the length of the CI. This quantity can be considered as a measure of precision of our estimate: given, say, two methods of interval estimation and a CL γ , the method leading to the smaller Dγ is to be preferred. Whichever the adopted method, however, it is reasonable to expect that there must be a relation between D and γ because – for a ﬁxed sample size n – a higher conﬁdence level (or, equivalently, a lower α) is paid at the price of a larger interval. In fact, choosing an unreasonably high value of γ generally leads to a CI which is too large to be of any practical use (and consequently to almost no information on θ). If we want a high CL and an interval of acceptable length we can, of course, increase the sample size. Since this operation is generally costly, it is evident that any procedure of interval estimation implicitly implies a compromise between conﬁdence level, interval length and sample size. (iii) Equation (5.69) deﬁnes a two-sided interval but in some applications one-sided intervals are required; these intervals have the form (−∞, T2 ) or (T1 , ∞). (iv) In case of several unknown parameters, the CI for an individual component, say θi , is still given by (5.69) and the same applies in case of a scalar function τ (q) of the unknown parameter(s). Clearly, θ is replaced by θi in the former case and by τ (q) in the latter. More specifically, a γ -conﬁdence region for the vector parameter q = (θ1 , . . . , θk ) is a random subset Cγ (X) ⊂ such that for all q ∈ we have Pq {q ∈ Cγ (X)} ≥ γ

(5.69c)

The general technique used to determine conﬁdence intervals is based on the search of a so-called pivot quantity. This is a r.v. of the form G(X; θ ) – that is, it depends on the sample and on the unknown parameter and therefore it is not a statistic – such that (1) its distribution fG does not depend on θ and (2) for every x the function G(x; θ ) is continuous and strictly monotone in θ. Then, given γ ∈ (0, 1) there are many ways in which we can choose g1 < g2 so that the relation Pθ {g1 < G(X; θ ) < g2 } =

g2

g1

fG (g) dg = γ

(5.70)

206 Mathematical statistics holds. If, for every x, we deﬁne T1 (x) and T2 (x) – with T1 < T2 – as the solutions (with respect to θ) of the equations G(x; θ ) = g1 and G(x; θ ) = g2 , respectively, eq. (5.70) is equivalent to eq. (5.69). Note that T1 , T2 are welldeﬁned because they are obtained by means of the inverse (with respect to θ) function G−1 , which, in turn, is well-deﬁned by virtue of condition (2). So, if G is monotonically increasing then T1 (x) = G−1 (x; g1 ) and T2 (x) = G−1 (x; g2 ) while, on the other hand, T1 (x) = G−1 (x; g2 ) and T2 (x) = G−1 (x; g1 ) if G is monotonically decreasing. The question at this point is how to construct a pivot quantity. A number of useful results given in Appendix C will be of help in this task (see also the following examples) but here we outline a general procedure. Suppose we are dealing with an absolutely continuous model; it can be shown that if the parent r.v. X has a PDF FX (x; θ ) which is continuous and strictly monotone in θ then G(X; θ ) = −

n

ln F(Xi ; θ )

(5.71)

i=1

is a pivot quantity for the interval estimation of θ. The proof, which we only outline here, is based on the fact if X has a continuous and monotonically increasing PDF F(x) then the chain of equalities FY (y) = P(Y ≤ y) = P{F(X) ≤ y} = P{X ≤ F −1 (y)} = F[F −1 (y)] = y shows that the r.v. Y ≡ F(X) has a uniform distribution on the interval (0, 1). Consequently, each r.v. F(Xi ; θ ) in (5.71) is uniformly distributed on (0, 1), − ln F(Xi ; θ ) has a (1, 1) distribution and G(X; θ ) has a (1, n) pdf, that is, fG (g) =

g n−1 e−g (n)

(5.72)

which does not depend on θ. Since G(X; θ ) is evidently continuous and monotone in θ, it follows that it is a pivot quantity. So, by taking (5.72) into account and choosing g1 , g2 such thateq. (5.70) holds, the solutions of the equations − ln F(xi ; θ ) = g1 and − ln F(xi ; θ ) = g2 give the desired conﬁdence interval. This last step, in practice, is often the most difﬁcult part. Before giving some examples, we mention the following useful result (whose proof is immediate): Proposition 5.17 If (T1 , T2 ) is a γ -CI for θ and h is a strictly monotone function, then h(T1 ) and h(T2 ) are the limits of the γ -CI for h(θ ). The interval is (h(T1 ), h(T2 )) if h is monotonically increasing and (h(T2 ), h(T1 )) if h is monotonically decreasing.

Preliminary ideas and notions

207

Example 5.9(a) Let X be a sample from a normal population with unknown mean µ = θ and known variance. In this case only a small effort is required √ to see that the r.v. G = n(M − θ )/σ is a pivot quantity (condition (1) above follows from the fact that G ≈ N(0, 1) – see Section 5.3.1, Proposition 5.1(b) – and therefore its pdf does not depend on θ). Consequently, our γ -CI has the form

g2 σ g1 σ (T1 , T2 ) = M − √ , M − √ (5.73) n n any two numbers such that g1 < g2 and (g2 ) − (g1 ) = γ where g1 , g2 are √ x 2 (where (x) = ( 2π )−1 −∞ e−t /2 dt is the PDF of a standard normal r.v.). The shortest interval can be obtained by minimizing the function σ Dγ (g1 , g2 ) = √ (g2 − g1 ) n

(5.74)

under the constraint (g2 ) − (g1 ) = γ (we note in passing that this is a rather rare case where Dγ does not depend on X). Using the well-known method of Lagrange undeterminate multipliers and taking into account that the standard normal pdf is an even function we get g1 = −g2 . Then, since (−x) = 1 − (x) it follows that (g1 ) = (1 − γ )/2 = 1 − (g2 ) and (g2 ) = (1 + γ )/2. By calling c(1+γ )/2 the (1 + γ )/2-quantile of the standard normal distribution, that is, c(1+γ )/2 = −1 [(1 + γ )/2] (this, in other words, is that particular value of g2 that minimizes the interval length) the desired γ -CI for the mean is

σ σ (T1 , T2 ) = M − c(1+γ )/2 √ , M + c(1+γ )/2 √ (5.75a) n n where the values of c(1+γ )/2 can be found in statistical tables. The interval length is in this case σ Dγ = 2c(1+γ )/2 √ n

(5.75b)

So, for instance, if γ = 0.95 then (1 + γ )/2 = 0.975 and we ﬁnd c0.975 = 1.960 while at a higher conﬁdence level, say γ = 0.99, we get (1 + γ )/2 = 0.995 and c0.995 = 2.576. As noted in point (ii) eq. (5.75b) shows that a higher CL, for a given sample size n, is paid at the price of a longer interval; for a given conﬁdence level, on the other hand, the interval length can only be reduced by increasing n. Suppose now that we had used the median Z instead of M. We have pointed out at the end of Section 5.3.1 normal with , that Z is asymptotically , mean µ and standard deviation σ π/2n, that is, r = π/2 times the standard deviation of M. If, just for the sake of the argument, we suppose that the

208 Mathematical statistics error of the approximation can be neglected (in other words, we pretend√ that the distribution of Z is exactly normal) we get the γ -CI (Z±c(1+γ )/2 rσ/2 n), which is longer than (5.75) although the risk of error is the same. Example 5.9(b) Consider now the (more frequent) case in which the vari2 (eq. (5.31)) is an unbiased estimator of σ 2 we ance is not known. Since S¯√ may think of using G = n(M − θ )/S¯ as a pivot quantity. In this case, however, it can be shown that G ≈ St(n − 1) and therefore the quantiles of the Student distribution (with n − 1 degrees of freedom) will have to be used in specifying our conﬁdence interval for the mean. The symmetry of the distribution suggests that we can parallel the considerations above on g1 , g2 and arrive at the CI S¯ S¯ (T1 , T2 ) = M − t(1+γ )/2;n−1 √ , M + t(1+γ )/2;n−1 √ (5.76) n n where, denoting by S(n−1) the Student PDF with n−1 degrees of freedom, we −1 [(1 + γ )/2]. The values of these quantiles are also have t(1+γ )/2;n−1 = S(n−1) easily found on statistical tables for ν (the number of degrees of freedom) up to 40–50. Tables for higher values of ν are not given because St(ν) → N(0, 1) as ν → ∞ and the normal approximation is already rather good for ν ≥ 30. ¯ and therefore the interNote that now Dγ depends on the sample (through S) val length is a r.v. which can only be determined after we have carried out our experiment. Nonetheless, also in this case we expect the considerations of point (ii) to hold. As a numerical example of cases (a) and (b) suppose that we test 20 similar products and obtain an average weight of M = 100.2 g. If we know that the population standard deviation is, say, σ = 4 g, the 95%-CI for M is (eq. (5.75))

4 4 100.2 − 1.96 √ , 100.2 + 1.96 √ 20 20

= (98.45, 101.95)

If, on the other hand, we make no assumptions on the variance and calculate it from the data obtaining, say, s¯ = 3.80 g, we use eq. (5.76) to get

3.8 3.8 100.2 − 2.093 √ , 100.2 + 2.093 √ 20 20

= (98.42, 101.98)

because for γ = 0.95, (1 + γ )/2 = 0.975 and we ﬁnd from the tables (for ν = 19) the quantile t0.975;19 = 2.093. Note that the second interval is larger than the ﬁrst even if the estimated standard deviation is smaller than the true σ . This situation may occur in practice because in the second

Preliminary ideas and notions

209

case the uncertainty on the standard deviation also plays a part. Moreover, if we carried out another experiment on other 20 items giving, by chance, M = 100.2, the ﬁrst interval would not change while the second will because ¯ of the new estimate of S. A further consideration on example (a) is that eq. (5.75b) gives us the possibility to determine the minimum sample size needed to achieve a speciﬁed ‘precision’ of our estimate at a given CL. In fact, if the ‘precision’ is measured by Dγ , there may be cases in which we do not want our CI to exceed √a given length L. This condition is expressed by the relation 2c(1+γ )/2 σ/ n ≤ L which can be solved for n to give n≥

2c(1+γ )/2 σ L

2 (5.77)

Example 5.10(a) Suppose that we are still dealing with a normal model; now, however, we know the mean µ and the variance is unknown. Setting θ = σ , this means that we are looking for a CI for the function τ (θ) = θ 2 . It is not difﬁcult to see that G(X; θ ) =

n 1 (Xi − µ)2 θ2

(5.78)

i=1

is a pivot quantity. Now, since (Xi − µ)/θ ≈ N(0, 1) it is known (Appendix C) that (Xi − µ)2 /θ 2 ≈ χ 2 (1) from which it follows that G(X; θ ) ≈ χ 2 (n) by the reproducibility property of the χ 2 distribution. Solving the equations G(x; θ ) = g1 and G(x; θ ) = g2 we get a CI of the form (T1 (X), T2 (X)) =

g2−1

(Xi − µ)

2

, g1−1

i

(Xi − µ)

2

(5.79)

i

where – denoting by Kn (x) the PDF of the distribution χ 2 (n) – g1 , g2 must satisfy the condition Kn (g2 ) − Kn (g1 ) = γ . A common choice is to select a so-called ‘central’ interval, that is, to choose g1 , g2 as the (1 ∓ γ )/2 quantiles of χ 2 (n), respectively. This gives 2 g1 = Kn−1 [((1 − γ )/2] = χ(1−γ )/2;n −1 2 g2 = Kn [((1 + γ )/2] = χ(1+γ )/2;n

(5.80)

so that the CI (5.79) is explicitly written as (T1 , T2 ) =

nS2

nS2

, 2 2 χ(1+γ )/2;n χ(1−γ )/2;n

(5.81)

210 Mathematical statistics and the values of the quantiles can be found on statistical tables. So, for instance, if we are looking for a 95%-CI and n = 20, then (1−γ )/2 = 0.025 2 and (1 + γ )/2 = 0.975. Since on tables of χ 2 quantiles we ﬁnd χ0.025;20 = 2 2 2 9.59 and χ0.975;20 = 34.17, our interval is (0.59S , 2.09S ). Two remarks on this example: ﬁrst, √ √ it is a direct consequence of Proposition 5.17 that the interval ( T1 , T2 ) – where T1 , T2 are as in (5.81) – is a γ -CI for the standard deviation σ . Second, using Lagrange’s method one can determine that the estimated interval (5.81) is not optimal, that is, is not the shortest one. A quantitative evaluation, however, is not immediate and requires a numerical solution. For a 95%-CI it can be shown that the shortest interval involves two quantities α1 , α2 such that α1 +α2 = 1−γ and the corresponding quantiles are 9.96 and 35.23 (instead of 9.59 and 34.17). Example 5.10(b) If, as it often happens, also the mean of the population is not known, a pivot quantity is given by (5.78) by simply substituting M in place of µ, that is, G(X; θ ) = (n − 1)S¯ 2 /θ 2 = (n − 1)S¯ 2 /τ (where, as above, τ (θ) = θ 2 ). In this case G(X; θ ) ≈ χ 2 (n − 1) and we get the CI for the variance n−1 n−1 2 2 ¯ ¯ (T1 , T2 ) = (5.82) S , 2 S 2 χ(1+γ χ(1−γ )/2;n−1 )/2;n−1 so that, for instance, for n = 20 and γ = 0.95 we ﬁnd in tables the two 2 2 2 2 quantiles χ(1+γ )/2;n−1 = χ0.975;19 = 32.85 and χ(1−γ )/2;n = χ0.025;19 = 8.907. As above, the central CI (5.81) is not the shortest interval but it is the most frequently used in practice. If, at this point we also want a CI for the mean, we proceed exactly as in Example 5.9(b) thus obtaining the interval (5.76) which – owing to the symmetry of the Student distribution – is the ¯ M + a2 S). ¯ shortest among all intervals of the form (M − a1 S, Example 5.11(a) From the preceding examples it appears that the determination of CIs for the (unknown) mean of a normal population involves (i) standardized normal quantiles if the variance is known or (ii) Student quantiles – with the appropriate number of degrees of freedom – if the variance is not known. Provided that collective independence of the r.v.s involved in the estimation problem applies, this is a general fact. Suppose in fact, that we want to ﬁnd a CI for the difference µ1 − µ2 where µ1 = θ1 , µ2 = θ2 are the means of two normal populations with variances σ12 , σ22 , respectively. Also, let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Ym ) be the samples taken from the two populations and M1 , M2 the two sample means. If the variances are known then we can exploit the fact that G=

M1 − M2 − (θ1 − θ2 ) ≈ N(0, 1) ) n−1 σ12 + m−1 σ22

(5.83)

Preliminary ideas and notions

211

and therefore G is a pivot quantity. Proceeding exactly as in Example 5.9(a) we obtain the CI :

⎛ ⎝M1 − M2 ± c(1+γ )/2

⎞ σ12 σ22 ⎠ + n m

(5.84)

If the variances are not known we use the estimators S¯ 12 , S¯ 22 (or S12 , S22 ) instead of the population variances. Using these estimators, it is convenient to introduce the ‘pooled’ variance Sp2 =

nS2 + mS22 (n − 1)S¯ 12 + (m − 1)S¯ 22 = 1 n+m−2 n+m−2

(5.85)

because it can be shown (Appendix C) that the r.v. G=

M1 − M2 − (θ1 − θ2 ) , Sp n−1 + m−1

(5.86)

is distributed as a Student variable with n + m − 2 degrees of freedom. This is our pivot quantity for the case at hand and we can parallel Example 5.9(b) to get the CI

, M1 − M2 ± t(1+γ )/2;n+m−2 Sp n−1 + m−1 9 m+n = M1 − M2 ± t(1+γ )/2;n+m−2 nS12 + mS22 mn(m + n − 2) (5.87)

where the second expression has been written in terms of the sample variances S12 , S22 . Example 5.11(b) As above, let X = (X1 , . . . , Xn ) and Y = (Y1 , . . . , Ym ) be independent samples from normal populations with unknown variances σ12 = θ12 , σ22 = θ22 , respectively. Now we wish to determine a CI for the ratio τ (θ1 , θ2 ) = θ12 /θ22 . The pivot quantity for this problem is obtained by noting that (Appendix C) Z1 = (n − 1)S¯ 12 /σ12 ≈ χ 2 (n − 1) and Z2 = (m − 1)S¯ 22 /σ22 ≈ χ 2 (m − 1) so that the r.v. S¯ 2 /θ 2 1 G = 12 12 = ¯S /θ τ 2 2

S¯ 12 S¯ 2 2

(5.88)

212 Mathematical statistics has a Fisher distribution with n − 1 and m − 1 degrees of freedom. Solving (5.88) for τ we get an interval of the form

S¯ 2 S¯ 2 g2−1 12 , g1−1 12 S¯ 2 S¯ 2

(5.89a)

so that denoting by F(1−γ )/2;n−1,m−1 and F(1+γ )/2;n−1,m−1 , respectively, the (1 − γ )/2 and (1 + γ )/2 quantiles of the distribution Fsh(n − 1, m − 1) the desired CI for the variance ratio is

S¯ 12 /S¯ 22

S¯ 12 /S¯ 22

, F(1+γ )/2;n−1,m−1 F(1−γ )/2;n−1,m−1 S¯ 12 S¯ 12 /S¯ 22 = , F(1+γ )/2;m−1,n−1 2 F(1+γ )/2;n−1,m−1 S¯ 2

(5.89b)

where in the second expression we took into account the property F(1−γ )/2;n−1,m−1 = {F(1+γ )/2;m−1,n−1 }−1 . So, for instance, if γ = 0.90, n = 20 and m = 15 we ﬁnd F0.95;19,14 = 2.40 and F0.95;14,19 = 2.26 and our interval is (0.417S¯ 12 /S¯ 22 , 2.26/S¯ 12 /S¯ 22 ). Example 5.11(c) As an example of a non-normal model, consider a sample taken from an exponential population with unknown mean (i.e. the statistical model is expressed in terms of the pdfs f (x; θ ) = θ −1 e−x/θ ). Now, since C) 2Xi /θ ≈ Exp(2) = χ 2 (2) and Xi ≈ Exp(θ) it follows that (Appendix −1 2 therefore G = 2θ i Xi ≈ χ (2n). It is left to the reader to ﬁll in the easy details and arrive at the central CI

2

Xi

2

Xi

, 2 2 χ(1+γ )/2;2n χ(1−γ )/2;2n

=

2nM

2nM

, 2 2 χ(1+γ )/2;2n χ(1−γ )/2;2n

(5.90)

2 As a numerical example, let γ = 0.90 and n = 10. We ﬁnd χ(1+γ )/2;2n =

2 2 2 = 31.41 and χ(1−γ χ0.95;20 )/2;2n = χ0.05;20 = 10.85; consequently (0.64M, 1.84M).

At this point a remark on notation is in order: whenever we have spoken of quantiles we meant lower quantiles. Some statistical tables report lower quantiles, but some other tables do not. In other words, if FG is the PDF under consideration (Gaussian, Student, χ 2 , Fisher, or else, depending on

Preliminary ideas and notions

213

the problem) and fG its density, our convention so far is that g1 FG (g1 ) =

fG (g) dg = (1 − γ )/2 = α/2 −∞

(5.91)

g2 FG (g2 ) =

fG (g) dg = (1 + γ )/2 = 1 − α/2 −∞

(we recall that γ = 1 − α by deﬁnition) so that the area under the pdf to the left of g1 equals α/2 and we can say, equivalently, that g1 is the α/2-lower quantile or, as we did, the (1 − γ )/2-lower quantile. Similarly, g2 is the (1 − α/2)-lower quantile or, equivalently, the (1 + γ )/2-lower quantile. In fact, for instance, one often ﬁnds – e.g. see [25] – the interval (5.81) written 2 2 as (nS2 /χ1−α/2;n , nS2 /χα/2;n ). From the ﬁrst of eq. (5.91), however, it follows that the area to the right ∞ of g1 is 1 − α/2, that is, P(G > g1 ) = g1 fG dg = 1 − α/2. Since the value of the area to the right of a given point is used to deﬁne the so-called ‘upper quantile’ of a distribution, the other convention sees g1 is the upper (1 − α/2)-upper quantile. By the same token, g2 is the upper α/2-upper quantile. Obviously, nothing changes for the degrees of freedom. So, for instance, one can ﬁnd eq. (5.81) written in terms of upper quantiles as (nS2 /χα/2;n , nS2 /χ1−α/2;n ) and now, if we look for a 95%-CI with, say, n = 20, we ﬁnd (see, for instance, Table 4 on [4] or Table C in Appendix II of 2 2 2 2 [7]) χα/2;n = χ0.025;20 = 34.17 and χ1−α/2;n = χ0.975;20 = 9.59. Obviously, the resulting interval is the same as above. In the following, in order to avoid confusion, we will explicitly state which type of quantile we are using; it must be the analyst’s care to check the tables at his/her disposal. Besides this observation on symbolism, it may also be worth spending a few words on some other interesting aspects of interval estimation. We start with the vector parameter case, which was brieﬂy mentioned in remark (iv) at the beginning of this section. The general technique used to construct conﬁdence regions is based on the fact that eq. (5.69c) is equivalent to Pq {X ∈ H(q)} ≥ γ

(5.92)

where, for every q ∈ , the set H(q) is the subset of the sample space containing all those realizations x (i.e. all those values taken on by X) such that the conﬁdence region constructed with these x will include q. So, the desired conﬁdence region is found by determining the sets H(q) satisfying inequality (5.92). Since, for a given CL, the sets H(q) can be chosen in many ways, the conﬁdence region thus constructed is not unique and the problem remains of ﬁnding a ‘minimal’ conﬁdence region. In practice, one generally ﬁnds the

214 Mathematical statistics sets H(q) with the help of some vector statistic T(X) with known distribution. As an example, we can reconsider Example 5.10(b) – normal model with unknown mean and variance – where we determined separate CIs for the mean and the variance. If, however, one considers the two-dimensional vector parameter q = (µ, σ 2 ) = (θ1 , τ ), it is wrong to deduce that the rectangle delimited by the intervals (5.76) and (5.82) is a γ -conﬁdence region for q. This is because the pivot quantities used to construct the CIs are related. Since it can be shown [19] that for a normal population the components of the two-dimensional statistic T = (M, S2 ) are independent, we can use the results (5.76) and (5.82) to obtain the set 3 , 4 H(q) = x : n/τ |m − θ1 | < a; b < ns2 /τ < b

(5.93)

2 2 where a = t(1+γ1 )/2;n−1 , b = χ(1−γ and b = χ(1+γ . Moreover, 2 )/2;n−1 2 )/2;n−1

the quantities γ1 , γ2 – owing to the independence of M and S2 – must satisfy the condition γ1 γ2 = γ in order to have a γ -conﬁdence region. Solving the inequalities which deﬁne H(q) we ﬁnd τ > n(m − θ1 )2 /a and ns2 /b < τ < ns2 /b . In the (θ1 , τ )-plane, therefore, the conﬁdence region is the part of the plane bounded by the parabola τ = n(m − θ1 )2 /a and the two straight lines τ = ns2 /b and τ = ns2 /b . Returning now to the one-dimensional case, a second consideration is the answer to the question: given a point estimator T(X) (of θ) with known distribution FT (t; θ ), can we construct a CI for θ? Intuitively, the answer is yes and, in fact, it is so. Let us assume that FT (t; θ ) is continuous and monotone in θ. Then, for every value of θ ∈ it is possible to deﬁne two numbers t1 , t2 (t1 < t2 ) such that Pθ {t1 < T(X) < t2 } = FT (t2 ; θ ) − FT (t1 ; θ ) = γ

(5.94)

Although they are not random quantities (because they are two realizations of T(X)) , t1 , t2 will be different for different values of θ; consequently, we can write t1 (θ ), t2 (θ ) and note that these two functions will generally be monotonically increasing in θ (if t is any sort of reasonable estimate of θ, it should increase as θ increases). Moreover, in order to uniquely deﬁne t1 , t2 one generally seeks a central interval by choosing them so that FT (t1 ; θ ) = (1 − γ )/2 FT (t2 ; θ ) = (1 + γ )/2

(5.95)

In the (θ, t)-plane we will therefore be able to identify a region bounded by the two functions t1 (θ ), t2 (θ ). This region, by construction, is such that eq. (5.94) holds for any ﬁxed value of θ ∈ ; but the important point is that for any ﬁxed value of t it deﬁnes two values θ1 (t), θ2 (t) – that is, the intersection of the horizontal line t with the curves t1 (θ ), t2 (θ ) – such that the interval (θ1 , θ2 ), in the long run, will bracket θ in γ % of the cases. This is

Preliminary ideas and notions

215

precisely the notion of conﬁdence interval for θ and therefore (T1 (X), T2 (X)), where Ti (X) = θi (T(X)) for i = 1, 2, is the desired γ -CI. So, under the assumptions above, we can in practice proceed as follows: given T(X) we obtain the realization x and consequently the estimate t = T(x); then, solving for θ the equations FT (t; θ ) = (1−γ )/2 and FT (t; θ ) = (1+γ )/2 we determine the extremes θ1 and θ2 of the γ -interval. By so doing, in the long run, we will be wrong (1 − γ )% of the times. We close this section with a ﬁnal observation on the examples above where, as the reader has probably noticed, we often assume a normal population as the starting statistical model. Although, clearly, the assumption of normality is not always justiﬁed in practice, we just point out two facts in its favour: (a) it has been shown that moderate and, sometimes, even signiﬁcant departures from normality lead to acceptable results in many cases and (b) if we suspect serious departures from normality, there is always the possibility of trying a transformation of the parent r.v. X (see, for instance, √ Ref. [2]) because log(X), X or some other function of it are often more nearly normal. Nonetheless, it goes without saying that in practical cases it is always advisable to check the basic assumption itself by carrying out a preliminary normality test on the data (this aspect is delayed to Chapter 6). 5.6.1

Asymptotic conﬁdence intervals

Consider√a point estimator Tn (X) of the unknown parameter θ such that the r.v. n(Tn − θ ) is asymptotically normal with zero mean and variance 2 2 σ √ (θ). If σ (θ) is a continuous function then it can be shown [19] that n(Tn − θ)/σ (Tn ) → N(0, 1) [D] as n → ∞. Consequently, for all θ we have √

n|Tn − θ| < c → (c) − (−c) = 2(c) − 1 = γ (5.96a) Pθ σ (Tn ) where c ≡ c(1+γ )/2 is the (1 + γ )/2-quantile of the standard normal distribution introduced in Example 5.9(a) and σ (Tn ) is the standard deviation of Tn . Since the relation above can be rewritten as Pθ

σ (Tn ) σ (Tn ) Tn − c(1+γ )/2 √ < θ < Tn + c(1+γ )/2 √ n n

→γ

(5.96b)

√ it follows that (Tn ± c1+γ /2 σ (Tn )/ n) is an asymptotic γ -CI for θ, where it is evident that the smaller is σ (Tn ) the shorter is the interval. As a consequence, asymptotically efﬁcient estimators will give the asymptotically shortest interval. If we recall from Section 5.5.1 that for regular models maximumlikelihood estimators are (i) asymptotically normal and (ii) asymptotically efﬁcient with variance 1/nI(θ ) = 1/In (θ ) – that is, the Cramer–Rao

216 Mathematical statistics limit – then the interval

c(1+γ )/2 Tˆ n ± √ nI(θ )

(5.97a)

(where Tˆ n is the ML estimator of θ) is the asymptotically shortest γ -CI for θ. Then, in order to ‘stablize’ the variance – that is, make it independent on θ – one may proceed as in Section 5.5.1 (eq. (5.66) and Example 5.8(a)) to obtain the conﬁdence interval for h(θ )

√ h(Tˆ n ) ± c(1+γ )/2 b/ n

(5.97b)

where, for simplicity, we chose a = 0 in eq. (5.66). If h is a monotone function we can then solve the resulting inequalities for θ to get the desired asymptotic γ -CI for the parameter θ. Owing to their nature, asymptotic CIs are exact only in the limit of n → ∞ but in common practice they are often used as approximate conﬁdence intervals when the sample is large – with the obvious understanding that the larger is the sample, the better is the approximation. As it should be expected, however, the notion of ‘large’ sample depends on the problem at hand because the rate of convergence to the normal distribution is not the same for all estimators. Nonetheless, it is a widely adopted rule of thumb that n > 30 can be considered a large sample when estimating conﬁdence intervals for means while n > 100 is the ‘dividing line’ between small and large samples when estimating conﬁdence intervals for variances. Example 5.12(a) In Example 5.8(a), we determined that the sample mean M is the ML estimator of the parameter θ of a Poisson model. Also, we found I(θ) √ = 1/θ noted that – choosing a = 0 and b = 1/2 in eq. (5.66) – the √ and √ r.v. n( M − θ ) is asymptotically normal √ with zero mean √ and variance 1/4. Then, it follows from eq. (5.97b) that ( M ± c(1+γ )/2 /2 n) is, for large √ samples, an approximate γ -CI for θ; consequently √ √ 2 √ √ 2 M − c(1+γ )/2 /2 n , M + c(1+γ )/2 /2 n

(5.98)

is the approximate γ -CI for θ. Example 5.12(b) For a sequence of n Bernoulli trials we have seen in Example 5.7(a) that the ML estimate of the parameter θ = p is the observed frequency of success x/n (which coincides with the sample mean M if 1 counts as a success and 0 counts as a failure). It is left to the reader to

Preliminary ideas and notions

217

show that I(θ) =

1 θ (1 − θ )

(5.99)

and therefore the approximate CI for θ is M±

c(1+γ )/2 , θ (1 − θ ) √ n

(5.100)

The stabilizing transformation can be obtained from eq. (5.66) which, setting a = 0 and b = 1/2, yields h(θ) =

1 2

√ dθ = arcsin( θ ) √ θ (1 − θ )

(5.101)

√ √ √ so that (arcsin( M) ± c(1+γ )/2 /2 n) is the approximate CI for arcsin θ.

5.7

A few notes on other types of statistical intervals

The somewhat detailed discussion of the preceding sections on conﬁdence intervals should not lead one to think that they are the only statistical intervals used in practice. Besides CIs, in fact, it is rather common in many applications to consider ‘tolerance intervals’ (TI) or ‘prediction intervals’ (PI), where the choice between the three types is dictated by the ﬁnal scope of the analysis. So, referring for the most part to Chapter 5 of [27], this section is simply meant to outline the main ideas behind these different concepts of statistical intervals. Before we do this, however, it is worth recalling that (a) the basic assumption is to draw a random sample from some population and (b) the statistical inferences are only valid for the population from which the sample was selected. In general, moreover, the assumption of normality is often made even if it may not be strictly met in practice. In this regard, the considerations at the end of Section 5.6 apply and in case of strong evidence of non-normality, one may always consider the possibility of using distribution-free methods (see, for instance, Ref. [13]). Tolerance intervals are needed when we are interested in an interval which will contain a certain percentage of the population. In this case, therefore, we will have two percentages: the percentage of population included in the interval and the conﬁdence level – often, as for CIs, 90, 95 or 99% – associated to the interval. This second percentage is usually included in the name and one speaks of 90%, 95% or 99%-TI, respectively. Assuming a sample from a normal population, tolerance intervals are ¯ and the values of cT,R – where the generally given in the form (M ± cT,R (n)S) subscript T is for ‘tolerance’ and R indicates the percentage of population contained in the interval – can be found in statistical tables for different

218 Mathematical statistics values of the sample size n. So, for instance, for n = 15 and a 95%-TI, we ﬁnd the values cT,90 = 2.48, cT,95 = 2.95 and cT,99 = 3.88. As the name itself implies, prediction intervals have to do with future observations. More speciﬁcally, a PI is needed when we are interested in an interval which will contain a speciﬁed number k of future observations from the population under study. So, for instance, given the population of daily ﬂights from, say, New York to Chicago, a pilot may not be interested in the average delay of these ﬂights, but in the delay of the next ﬂight in which he/she will be ﬂying. Similarly, a customer purchasing a small number of units of a given product is not interested in the long-run performance of the process from which his/her units are a sample, but in the quality of those particular units that he is buying. As for the other types of intervals, we associate to a PI a conﬁdence level but now the second deﬁning number is k, the number of future observations to be ¯ included in the interval. Again, the interval is given in the form (M±cP,k (n)S) where the subscript P is for ‘prediction’ and the values of cP,k can be found in statistical tables. As a numerical example, suppose that we have n = 10 observations from a normal population and we are interested in the values of k = 2 further randomly selected observations from that population. For n = 10, at a 95% CI we ﬁnd the value cP,2 = 2.79 so that our 95%-PI is ¯ where M and S¯ are the mean and (unbiased) standard deviation (M ± 2.79S), calculated on the basis of the ten observations at our disposal. An important difference between the types of intervals is that CIs become smaller and smaller as the sample size increases while it is not so for TIs and PIs. Finally, it is worth noting that there exist other types of prediction intervals such as, for instance, the PI to contain – at a given conﬁdence level – the mean of k future observations or the standard deviation of k future observations. For more detailed information the interested reader can refer to [13 and 14].

5.8

Summary and comments

The theory of Probability is an elegant and elaborate construction well worthy of study in its own right. Statistics, broadly speaking, is the other face of the coin because it provides the methods and techniques by which – on the basis of a limited number of observed data – we can make (inductive) inferences and/or draw conclusions on speciﬁc real-word problems where randomness is involved. In other words, one can safely say that Statistics ‘sees these problems from a different angle’, although it is evident that it must necessarily rely on Probability theory in order to be effective. The approach of Statistics is explained in Section 5.2, where the concept of statistical model is introduced together with the deﬁnitions of ‘sample’, ‘realization of the sample’ and some notes on the important aspect of data collection. With Section 5.3 we turn to more practical considerations by noting that one of the ﬁrst step in every analysis is to use the experimental data to

Preliminary ideas and notions

219

calculate the so-called ‘sample characteristics’ where, by analogy, each one of them is generally the counterpart of a well-deﬁned probabilistic quantity. In this light, therefore, one speaks of sample mean, sample variance, kth order (ordinary and central) sample moment, etc., and of their realizations which, in turn, may change from experiment to experiment because the realization of the sample, as a matter of fact, does change from experiment to experiment. Being random variables themselves, moreover, it makes sense to speak of mean, variance, etc. – and, more generally, of the probability distribution – of sample characteristics. All these aspects are discussed in Section 5.3 by implicitly assuming the sample size n as ﬁxed. This, however, is not the whole story because another important issue is considered in Section 5.3.1: the behaviour of sample characteristics as the sample size increases indeﬁnitely – that is, mathematically speaking, as n → ∞. In the limit, in fact, some important properties of both theoretical and practical interest show up: theoretical because an inﬁnite sample is an evident impossibility and consequently these asymptotic properties can never be realized in full, but practical because it can often be assumed that they are, to a certain extent, satisﬁed by large samples, thereby providing useful working approximations in many cases. Having introduced the concept of sample characteristic and, in particular, of statistic – that is, a sample characteristic containing no unknown quantities – both Sections 5.4 and 5.5 and all their subsections are dedicated to the subject of point estimation. In essence, the problem consists in estimating one or more unknown parameters of a supposedly known type of distribution by means of an appropriate statistic. The type of distribution provides the underlying statistical model while the observed data are used to calculate the ‘appropriate’ statistic which, we hope, will estimate the unknown parameter(s) within an acceptable degree of accuracy. Since this kind of problem is fundamental in almost all statistical applications, the ﬁrst step is to specify some criteria by which we may be able to decide whether a given statistic can qualify as a ‘good’ – or even, if and when possible, as the ‘best’ – estimator for the parameter under investigation. In this respect, in fact, it is not sufﬁcient to rely solely on analogy – that is, using the sample mean to estimate the mean, the sample variance for the variance, etc. – because it can be shown that this intuitive approach, although useful in some cases, may even be misleading in some other cases. Among the most important criteria to judge an estimator, Section 5.4 considers unbiasedness, asymptotic unbiasedness, efﬁciency and consistency. Then, in regard to efﬁciency, Section 5.4.1 deals with a fundamental result applying to the so-called regular problems: this is the Cramer–Rao inequality which, by establishing a lower limit for the variance of an estimator, can indicate the best estimator – when it exists – in terms of efﬁciency. In the process, the deﬁnition of Fisher’s information is given and all the concepts above are generalized to the case of a k-dimensional (vector) parameter and to a scalar function of a vector parameter.

220 Mathematical statistics Another desirable property of estimators is sufﬁciency. The deﬁnition is not self-evident and, often, is also of little practical use for the purpose of identifying sufﬁcient estimators. The required explanations are given in Section 5.4.2, where it is also shown that Neyman’s factorization theorem provides an easier way to assess sufﬁciency and that – Rao–Blacwell theorem – the so-called MVUE (minimum variance unbiased estimator), when it exists, is a function of a sufﬁcient statistic. The property of completeness, moreover, is introduced in order to state Lehmann–Scheffé theorem which, in turn, speciﬁes the general form of a MVUE as a function of a sufﬁcient and complete statistic and an unbiased estimator (for the unknown parameter under study). Finally, the way in which we can ﬁnd estimators with the above properties – or at least some of them – is explained in Section 5.5. Although not the only one, the most popular technique for this purpose is the so-called ML method. The name itself is self-explanatory and consists in maximizing the likelihood function (or, more often, its natural logarithm) with respect to the unknown parameter(s). The ‘method of moments’ and ‘Bayes’ method’ are also brieﬂy considered in Section 5.5 but it is noted that, in general, ML estimators have a number of desirable properties and here, probably, lies the reason for the method’s popularity. Particularly worthy of mention are the asymptotic properties of ML estimators considered in Section 5.5.1. The most appropriate solution to many problems is not in the form of a point estimate because the main concern is often a range of values within which we can conﬁdently hope to ﬁnd the true value of the unknown parameter. This is a so-called problem of interval estimation and is the subject of Section 5.6. So, by ﬁrst specifying a conﬁdence level γ , our goal is to determine two statistics T1 , T2 such that eq. (5.69) holds; these statistics, once we ﬁnd them, are the lower and upper limit of the CI, respectively. At this point we use the experimental data to calculate their realizations t1 , t2 and say that (t1 , t2 ) is the desired γ -CI. The general technique by which the task of ﬁnding T1 , T2 is accomplished is explained in Section 5.6 and the many worked-out examples show that conﬁdence intervals are always speciﬁed in terms of quantiles of an appropriate distribution where, on the one hand, the ‘appropriate’ distribution (frequently the Gaussian, the χ 2 or the Fisher distribution) depend on the parameter under study while, on the other hand, the quantiles to be used in actually calculating the interval depend on the conﬁdence level. In any case, however, it is pointed out that we cannot say that the true value θ of the parameter lies in the interval (t1 , t2 ) with probability θ. This is because (t1 , t2 ) is a ‘deterministic’ interval with nothing random in it and therefore θ either belongs to it or it does not. The correct statement is given in terms of the long-run interpretation of conﬁdence intervals: by repeating the estimation procedure many times – thus obtaining many conﬁdence intervals – θ will fall in these intervals in 100γ % of the cases. Also, another general fact is that the procedure of interval estimation must be based on a compromise

Preliminary ideas and notions

221

between sample size and conﬁdence level. For a given sample size, in fact, a higher conﬁdence level corresponds to a longer interval and therefore an unreasonably high value of γ will lead to an interval which may be too large to be of any practical use. The interval length, on the other hand, can be decreased by either choosing a lower conﬁdence level or by increasing the sample size, or both. Increasing the sample size, however, is generally costly and, in some cases, may not even be practicable. So, a correct balance of these quantities must be agreed upon at the planning stage and, clearly, it is the analyst’s responsibility – depending on the importance of the problem at hand – to suggest a viable solution. Finally, it is noted that cases in which ﬁnding a conﬁdence interval turns out to be a very difﬁcult task are not rare. For large samples, however, a practical solution is the use of asymptotic conﬁdence intervals and this is the subject of Section 5.6.1. In Section 5.7, moreover, we brieﬂy introduce the concepts of ‘tolerance intervals’ and ‘prediction intervals’ by also giving a number of speciﬁc references for the reader interested in more details on these further aspects of interval estimation.

References and further reading [1] Azzalini, A., ‘Inferenza Statistica: una Presentazione Basata sul Concetto di Verosimiglianza’, 2nd edn., Springer-Verlag Italia, Milano (2001). [2] Bartlett M.S., ‘The Use of Transformations’, Biometrics, 3, pp. 39–52 (1947). [3] Cramér, H., ‘Mathematical Methods of Statistics’, 19th edn., Princeton Univ. Press, Princeton (1999). [4] Crow, E.L., Davis, F.A., Maxﬁeld, M.W., ‘Statistics Manual’, Dover, New York (1960). [5] de Haan, L., ‘Sample extremes: an Elementary Introduction’, Stat. Neerlandica, 30, 161–172 (1976). [6] Di Crescenzo, A., Ricciardi, L.M., ‘Elementi di Statistica’, Liguori Editore, Napoli (2000). [7] Duncan, A.J., ‘Quality Control and Industrial Statistics’, 5th edn., Irwin, Homewood, Illinois (1986). [8] Edwards, A.W.F., ‘Likelihood’, The Johns Hopkins University Press, Baltimore (1992). [9] Fisher, R.A., ‘On the Mathematical Foundations of Theoretical Statistics’, PTRS, 222 (1921). [10] Galambos, J., ‘The Asymptotic Theory of Extreme Order Statistics’, 2nd edn., Krieger, Malabar (1987). [11] Gnedenko, B.V., ‘Sur la Distribution Limite du Terme Maximum d’une Série Aléatoire’, Ann. Math., 44, 423–453 (1943). [12] Green, J.R., Margerison, D., ‘Statistical Treatment of Experimental Data’, Elsevier, Amsterdam (1977). [13] Hahn, G.J., Meeker, W.Q., ‘Statistical Intervals: a Guide for Pratictioners’, Wiley, New York (1990). [14] Hahn, G.J., ‘Statistical Intervals for a Normal Population. Part I. Tables, Examples and Applications’, Journal of Quality Technology, July, 115–125

222 Mathematical statistics

[15] [16] [17] [18] [19] [20] [21] [22]

[23] [24] [25] [26] [27] [28] [29] [30]

(1970); ‘Part II. Formulas, Assumptions, some Derivations’, Journal of Quality Technology, October, 195–206 (1970). Huff, D., ‘How to Lie With Statistics’, W. W. Norton & Company, New York (1954). Keeping, E.S., ‘Introduction to Statistical Inference’, Dover, New York (1995). Klimov, G., ‘Probability Theory and Mathematical Statistics’ Mir Publishers, Mosow (1986). Kottegoda, N.T., Rosso, R., ‘Statistics, Probability and Reliability for Civil and Environmental Engineers’, McGraw-Hill, New York (1998). Ivchenko, G., Medvedev, Yu., ‘Mathematical Statistics’, Mir Publishers, Moscow (1990). Mandel, J., ‘The Statistical Analysis of Experimental Data’, Dover, New York, (1984). Mendenhall, W., Wackerly, D.D., Scheaffer, R.L., ‘Mathematical Statistics with Applications’, 4th end., PWS-KENT Publishing Company, Boston (1990). Nasri-Roudsari, D., Cramer, E., ‘On the Convergence Rates of Extreme Generalized Order Statistics’ www.math.uni-oldenburg.de/preprints/get/source/ Rates.pdf Pace, L., Salvan, A., ‘Teoria della Statistica’, CEDAM, Padova (1996) Rao, C.R., ‘Asymptotic Efﬁciency and Information’, Proc. 4th Berkeley Symp. Math. Stat. Prob. 1, 531–545 (1961). Rinne, H., ‘Taschenbuch der Statistik’, Verlag Harri Deutsch, Frankfurt am Main (2003). Serﬂing, R.J., ‘Approximation Theorems in Mathematical Statistics’, John Wiley & Sons, New York (1980). Wadsworth, H.M. (editor), ‘Handbook of Statistical Methods for Engineers and Scientists’, McGraw-Hill, New York (1990). Zacks, S., ‘The Theory of Statistical Inference’, John Wiley & Sons, New York (1971). Thisted, R.A., ‘Elements of Statistical Computing’, Chapman & Hall, London (1988). Buczolich Z., Székely G.J., Adv. Appl. Math., 10, 439–256 (1992).

6

6.1

The test of statistical hypotheses

Introduction

Broadly speaking, any assumption on the distribution of one or more random variables observed in an experiment is a statistical hypothesis. The hypothesis may be based on theoretical considerations, on the analysis of other (similar) experiments or it may just be an educated guess suggested by reasonableness or common sense, whatever these terms mean. In any case, it must be checked by actually performing the experiment and by devising some method which – in the light of the acquired data – gives us the possibility to decide whether to accept it or reject it. This, it should be clear from the outset, does not imply that our decision will be right because, as in any procedure of statistical inference, the best we can do (unless we examine the entire population) is to reduce the probability of being wrong to an acceptable level, where the term ‘acceptable’ generally depends on the problem at hand, the seriousness of the consequences of being wrong and, last but not least, the cost of the experiment. Consequently, we will not state our conclusions by saying ‘our hypothesis is true (false)’ but ‘the observed data are in favour (against) our hypothesis’, and we will continue our work behaving as if the hypothesis were true (false). The methods by means of which we make our decision are called statistical tests and are the subject of this chapter. We will ﬁrst illustrate the main ideas from a general point of view and then turn to typical classes of problems and speciﬁc examples.

6.2

General principles of hypotheses testing

Let us start with some deﬁnitions. The hypothesis to be tested, generally denoted by H0 , is called the null hypothesis and it is tested against an alternative hypothesis H1 . The two hypotheses are regarded as mutually exclusive and exhaustive. This is to say that if we accept H0 then we reject H1 and conversely, but it does not mean that – given H0 – the hypothesis H1 is the one and only alternative to H0 . As a matter of fact, it is often possible to conceive of several alternatives to H0 , say H1 , H1 and so on, but the

224 Mathematical statistics point is that once H0 and H1 have been formulated, the test leading to the acceptance/rejection of H0 necessarily leads to the rejection/acceptance of H1 . Clearly, it is the analyst’s responsibility to select the most appropriate pair of hypotheses for the problem at hand. So, given H0 and H1 , the experimental data form the evidence on the basis of which we decide to accept or reject H0 . Due to the intrinsic uncertainty of any statistical inference, our decision may be right or wrong; however, we may be wrong in two ways: (a) by rejecting H0 when in fact it is true; or (b) by accepting H0 when in fact it is false. The common terminology deﬁnes (a) a type I error (or rejecting error) and (b) a type II error (or acceptance error). Ideally, one would like both possibilities of error to be as small as possible, but since it turns out that, for a ﬁxed sample size n, it is generally not possible to decrease one type without increasing the other, some sort of compromising strategy must be adopted. We will come to this point shortly. In essence, any statistical test is a rule by which a realization x = (x1 , . . . , xn ) of the sample X = (X1 , . . . , Xn ) is used to make a decision about the assumption H0 . More speciﬁcally, this is done by dividing the sample space into two disjoint sets 0 , 1 – called the acceptance region and the rejection (or critical) region, respectively – such that 0 ∪ 1 = . As the names themselves imply, 0 contains all x which lead to the acceptance of H0 while 1 contains all x which lead to the rejection of the null hypothesis. In this light, the basic formulation of a statistical test is as follows: Let x be a realization of the sample X. If x ∈ 0 we accept the null hypothesis H0 ; if, on the other hand, x ∈ 1 we reject H0 (and therefore accept H1 ). Then, the two possibilities of error correspond to the cases: (a) x ∈ 1 when H0 is true and (b) x ∈ 0 when H0 is false. The selection of the acceptance and rejection regions is strictly related to two other aspects: the test chosen for a given null hypothesis and the ‘goodness’ of the test. In fact, since it is reasonable to expect that a given null hypothesis H0 can be tested by different methods and that each method will deﬁne its acceptance and rejection regions, the problem arises of which test to choose among all possible tests on H0 . The choice, we will see, depends also on the alternative hypothesis H1 but for the moment we assume both H0 and H1 as given. Now, an intuitive solution to this problem is, for a speciﬁed sample size n, to call ‘best’ the test which makes the possibility of error as small as possible and choose this one. This aspect, however, deserves further consideration because – keeping in mind that we do not know if H0 is true or not – the two types of error must be considered simultaneously. If, as it is customary, we denote by α and β the probabilities of committing a type I and type II error respectively, it turns out that we cannot simultaneously make them as small as we wish. This fact is evident if we examine

The test of statistical hypotheses

225

the two extreme cases. If we choose α = 0 we will never make a type I error and this, in turn, means that = 0 because we will always accept H0 regardless of the observed realization x. This is a correct decision if H0 is true (which we do not know); if, however, H0 is false, our choice of accepting it no matter what – since 1 = C 0 = ∅ – implies β = 1. Conversely, choosing β = 0 means that = 1 and 0 = ∅; therefore we will always reject H0 , a circumstance which prevents us from committing a type II error if H0 is false, but implies α = 1 if H0 is true. Between the two extremes there are many possible intermediate cases corresponding to different choices of 0 and 1 but it is a general fact that reducing α tends to increase β and viceversa. The usually adopted strategy to overcome this difﬁculty is due to Neyman and Pearson and is based on the consideration that in most cases one type of error has more serious consequences than the other. Consequently, we ﬁx a value for the probability of the worst error and, among all possible tests, we choose the one that minimizes the probability of the other error. Since the problem is often formulated in such a way that the type I error is the worst, the strategy consists in specifying a value for α and – if and when possible – choosing the test with the smallest value of β (or, equivalently, the maximum value of 1 − β) compatible with the prescribed value of α. This speciﬁed value of α – which, clearly, depends on practical considerations about the problem at hand – deﬁnes the signiﬁcance level of the test. Before turning to other general aspects of hypothesis testing, we open a short parenthesis on notation. Often one denotes the probabilities of type I and type II errors by P(H1 |H0 ) and P(H0 |H1 ), respectively. This symbolism does not mean that we are dealing with conditional probabilities in the strict sense, but it is just a convenient way of indicating – in the two cases – the accepted hypothesis (in the ﬁrst ‘slot’ within parenthesis) and the true hypothesis (in the second ‘slot’). Returning to the main discussion, an observation of practical nature is that the critical (rejection) region is frequently deﬁned by means of a so-called test function T(X), where T(X) is a statistic which must be appropriately chosen for the problem at hand. Having chosen a test statistic, the critical region will then be expressed in one of the following forms ⎧ ⎪ ⎨{x : T(x) ≥ c} 1 = {x : T(x) ≤ c} ⎪ ⎩ {x : |T(x)| ≥ c}

(6.1)

where c is a real number which depends on the signiﬁcance level α. This, in other words, means that for every α the set T = {t : t = T(x), x ∈ } of all possible values of T is divided into two subsets T0 , T1 , where T1 will include all those realizations t = T(x) which lead to the rejection of H0 .

226 Mathematical statistics A second comment worthy of mention is that, for a ﬁxed sample size n, some problems of hypothesis testing do not lend themselves easily to a solution. Things, however, often get better if we adopt an asymptotic approach by letting the sample size tend to inﬁnity. This is a frequently adopted strategy but it should be kept in mind that the ﬁnal results are then valid only for large samples. For moderate sample sizes, however, they can often be considered as useful working approximations. Having outlined the general philosophy of the statistical testing, it can be of help at this point to have an idea of some typical of types of hypotheses encountered in practice. The following is a short list: (1) Hypothesis on the form of distribution: In this case we make n independent observations of a r.v. X with unknown distribution FX (x) and use the acquired data x to check if the distribution of X is, as we assume, F(x). The null hypothesis is then written H0 : FX (x) = F(x). The function F(x), in turn, may be (a) completely deﬁned or (b) may belong to a certain class – for example, normal, Poisson, or else – the uncertainty being on one (or more) parameter(s) θ of the distribution. An example of this latter type can be H0 : FX (x) = N(µ, θ ), meaning that we want to test the hypothesis that X has a normal distribution with known mean µ and unknown variance θ. (2) Hypothesis of independence: In this case we have, for example, a twodimensional r.v. X = (X, Y) with unknown PDF FX (x, y) and we have reasons to believe that X and Y are independent. Then, the null hypothesis is symbolically expressed as H0 : FX (x, y) = FX (x)FY (y). (3) Hypothesis of homogeneity: We carry out a series of m independent experiments – each experiment consisting of n trials – obtaining the results (x1i , . . . , xni ), where i = 1, . . . , m. Our basic assumption in this case is that these data are homogeneous, that is, they are all observations of the same random variable. Then, since the null hypothesis is that the distribution law is the same for all the experiments, we symbolically express the problem as H0 : F1 (x) = F2 (x) = · · · = Fm (x), where we denoted by Fi (x) the (unknown) distribution of the ith experiment. Clearly, the types of hypothesis considered above do not cover all the possibilities because the list has been given mainly for illustrative purposes. Other speciﬁc cases will be examined in due time if and whenever needed in the course of future discussions.

6.3

Parametric hypotheses

If the hypothesis to be tested concerns one or more unknown parameters of a supposedly known type of probability distribution, one speaks of parametric hypotheses. The basic procedure is similar to what has been done in Chapter 5 – that is, we start from the statistical model (5.1) and, on the

The test of statistical hypotheses

227

basis of the acquired data, draw inferences on θ – but the details differ. In Chapter 5, in fact, we did not formulate any hypothesis whatsoever on θ and our main concern was simply to determine a reliable estimate of it, either in the form of a numerical value or a conﬁdence interval. Now we do formulate an hypothesis – the null hypothesis H0 – and the scope is to accept it or reject it depending on whether H0 is reasonably consistent with the observed data or not. This kind of approach is generally more convenient when, following the experiment, we must make a ‘yes or no’ decision and take action accordingly. For the moment we ignore the fact that the two problems – parametric hypothesis testing and conﬁdence interval estimation – are, in fact, related and delay the discussion of this aspect to Section 6.3.4. Denoting, as in Chapter 5, the parameter space by , the general form of the null and alternative hypotheses is H0 : θ ∈ 0 H1 : θ ∈ 1

(6.2)

where 0 , 1 are two subsets of such that 0 ∩ 1 = ∅ and 0 ∪ 1 = . More speciﬁcally, we call simple any hypothesis which speciﬁes the probability distribution completely, otherwise we speak of composite (or compound) hypothesis. So, for instance, H0 : θ = θ0 and H1 : θ = θ1 (where θ0 and θ1 are given numerical values) are simple hypotheses while H0 : θ ≥ θ0 , H1 : θ = θ0 or, say, H1 : θ < θ0 are composite hypotheses. Depending on the problem at hand, we may have any one of the three possibilities (i) both the null and alternative hypotheses are simple, (ii) one is simple and the other is composite and (iii) both hypotheses are composite. Before examining the various cases, we must return for a moment to the discussion of Section 6.2 on how to select a ‘good’ test, a choice which – we recall – requires a closer look at the two types of error. In case of parametric hypotheses, they generally depend on θ and can be written as α(θ) = Pθ (X ∈ 1 | θ ∈ 0 ) β(θ ) = Pθ (X ∈ 0 | θ ∈ 1 )

(6.3)

If we deﬁne the so-called power function W(θ ) as W(θ) =

Pθ (X ∈ 1 | θ ∈ 0 ) = α(θ) Pθ (X ∈ 1 | θ ∈ 1 ) = 1 − β(θ )

(6.4)

we recognize 1 − β as the probability of not making a type II error. Since an ideal test will result in W(θ ) = 0 if H0 is true (i.e. θ ∈ 0 ) and W(θ ) = 1 if H0 is false (i.e. θ ∈ 1 ), the function W can be used to compare different tests on a given pair of hypothesis H0 , H1 . In this light, in fact, we have the

228 Mathematical statistics following deﬁnitions: (i) we call size of a test the quantity α = sup W(θ )

(6.5)

θ∈0

(note that for parametric hypotheses the terms ‘size’ and ‘signiﬁcance level’ are interchangeable). (ii) given a test T on a pair of hypotheses H0 , H1 , let α be its size and β(θ ) its probability of a type II error. Then T is called the uniformly most powerful test if, for any other test T ∗ (on H0 , H1 ) of size α ∗ ≤ α, we have β ∗ (θ) ≥ β(θ ) for all θ ∈ 1 . This, in other words, means that the uniformly most powerful test T – denoting by W(θ ) its power function – satisﬁes the inequality W(θ) ≥ W ∗ (θ )

for all θ ∈ 1

(6.6)

Also, a desirable property for a test is unbiasedness. A test T is called unbiased if W(θ) ≥ α

for all θ ∈ 1

(6.7)

so that we have a higher probability of rejecting H0 when it is false than rejecting it when it is true. At this point, another word of caution is in order because mistakes and misunderstandings are rather frequent: the power function considers the probabilities of rejecting H0 when it is true and when it is false. This is, in essence, the main idea of hypothesis testing and nothing can be said about the probability of H0 being true or false. So, if H0 is accepted at, say, the 5% signiﬁcance level it does not mean that the probability of H0 being true is 95%. This distinction, as a matter of fact, is fundamental and should always be kept in mind when reporting the results. 6.3.1

Simple hypotheses: Neyman–Pearson’s lemma

A uniform more powerful test does not always exist because uniformity, that is, the condition ‘for all θ ∈ 1 ’, is rather strong and we may have cases in which two tests, say T1 , T2 , cannot be compared because W1 (θ ) < W2 (θ ) for some values of θ in 1 while W1 (θ ) > W2 (θ ) for some other values of θ in 1 . A most powerful test, however, always exists when we are dealing with a pair of simple hypothesis, that is, the case in which eq. (6.2) have

The test of statistical hypotheses

229

the form H0 : θ = θ0

(6.8)

H1 : θ = θ1

where θ0 , θ1 are two speciﬁc numerical values for the unknown parameter. Equation (6.8), in other words, imply that the parameter space consists of only two points – that is, = {θ0 , θ1 } – and that the distribution of the r.v. X is either F0 (x) = F(x; θ0 ) or F1 (x) = F(x; θ1 ), where F is a known type of PDF (normal, exponential, Poisson or else). Assuming that F0 and F1 are both absolutely continuous with densities f0 (x) and f1 (x), respectively (with f0 , f1 > 0), the following theorem – known as Neyman–Pearson’s lemma – holds. Proposition 6.1 (Neyman–Pearson’s lemma) Let (6.8) be the null and alternative hypotheses and x = (x1 , . . . , xn ) be a realization of the sample X = (X1 , . . . , Xn ). The most powerful test of size α is speciﬁed by the critical region 1 = {x : l(x) ≤ c}

(6.9)

where c (c ≥ 0) is such that Pθ0 [l(X) ≤ c] = α and l(X) is a statistic called the ‘likelihood-ratio’ and deﬁned as l(X) ≡

& f0 (Xi ) L(X; θ0 ) = &i L(X; θ1 ) i f1 (Xi )

(6.10)

In order to simplify the notation in the proof of the theorem let us call A the rejection region (6.9) and let B be the rejection region of another test of size α (on the hypotheses (6.8)). Then

L(x; θ0 ) dx =

A

L(x; θ0 ) dx = α B

because both tests have size α. Noting that both A and B can be expressed as the union of two disjoint sets by writing A = (A ∩ B) ∪ (A ∩ BC ) and B = (A ∩ B) ∪ (B ∩ AC ) respectively, the equality above implies

L(x; θ0 ) dx =

A∩BC

L(x; θ0 ) dx

(6.11)

B∩AC

By the deﬁnition of A, moreover, it follows that L(x; θ1 ) ≥ L(x; θ0 )/c for x ∈ A and, clearly, L(x; θ1 ) < L(x; θ0 )/c for x ∈ AC . Using these inequalities,

230 Mathematical statistics eq. (6.11) leads to the chain of relations

L(x; θ1 ) dx ≥

A∩BC

L(x; θ0 ) dx c

A∩BC

=

L(x; θ0 ) dx > c

L(x; θ1 ) dx B∩AC

B∩AC

which, in turn, are used to get

L(x; θ1 ) dx = A

L(x; θ1 ) dx + A∩B

>

L(x; θ1 ) dx

A∩BC

L(x; θ1 ) dx + A∩B

=

L(x; θ1 ) dx

B∩AC

L(x; θ1 ) dx

(6.12)

B

meaning that the probability 1 − β is higher for the test with rejection region A = 1 . In fact, since the quantity 1 − β (i.e. the probability of rejecting H0 when H0 is false or, equivalently, of accepting H1 when H1 is true) of a given test is obtained by integrating L(x; θ1 ) over its rejection region, eq. (6.12) proves the theorem because the test corresponding to B is any test of size α on the hypotheses (6.8). In addition, we can show that the test is always unbiased. In fact, in the rejection region A = 1 (eq. (6.9)) we have L(x; θ0 ) ≤ cL(x; θ1 ) which, if c ≤ 1, implies L(x; θ0 ) ≤ L(x; θ1 ) and therefore α=

L(x; θ1 ) dx = 1 − β = W(θ1 )

L(x; θ0 ) dx ≤ A

A

On the other hand, in the acceptance region AC we have L(x; θ0 ) > cL(x; θ1 ) and therefore L(x; θ0 ) > L(x; θ1 ) whenever c > 1. Consequently 1−α =

L(x; θ0 ) dx >

AC

L(x; θ1 ) dx = β = 1 − W(θ1 ) AC

thus showing that condition (6.7) holds in any case. Example 6.1(a) As an application of Neyman–Pearson’s lemma, consider a normal r.v. with known variance σ 2 . On the basis of the random sample X

The test of statistical hypotheses

231

and the observed data x we want to test the pair of simple hypotheses (6.8) on the unknown mean µ = θ where, for deﬁniteness, we assume θ1 > θ0 . We have n 1 [(xi − θ0 )2 − (xi − θ1 )2 ] l(x) = exp − 2 2σ i=1 nm n θ12 − θ02 − 2 (θ1 − θ0 ) = exp 2 2σ σ where m = xi is the realization of the sample mean calculated from the data. The inequality deﬁning the rejection region (6.9) holds if

m≥

(θ1 + θ0 ) σ 2 log c − 2 n(θ1 − θ0 )

(6.13a)

or, equivalently, if √ √ σ log c n(m − θ0 ) n(θ1 − θ0 ) ≥ −√ ≡ t(c) σ 2σ n(θ1 − θ0 )

(6.13b)

√ Now, noting that (Proposition 5.1(b)) the r.v. Z = n(M−θ0 )/σ is standard normal if H0 is true, we have Pθ0 [l(X) ≤ c] = Pθ0 [Z ≥ t(c)] = α and therefore t(c) is the α-upper quantile of the standard normal distribution. This quantity is found on statistical tables and is frequently denoted by the special symbol zα (we ﬁnd, for instance, for α = 0.05; 0.025; 0.01 – the most commonly adopted values of α – the upper quantiles z0.05 = 1.645, z0.025 = 1.960 and z0.01 = 2.326, respectively). Then, in agreement with Neyman–Pearson’s lemma, it follows that the most powerful test for our hypotheses is deﬁned by the critical region '

σ 1 = x : m ≥ θ0 + zα √ n

( (6.14a)

√ and its power is W(θ1 ) = 1 − β = Pθ1 {M ≥ θ0 + zα σ/ n}. This quantity √ can be obtained by noting that under the alternative hypothesis the r.v. n(M − θ1 )/σ is standard normal. Consequently, W(θ1 ) equals√the r-upper quantile of the standard normal distribution, where r = zα − n(θ1 − θ0 )/σ . Since r < zα the area (under the standard normal pdf) to the right of r is greater than the area to the right of zα – which, by deﬁnition, equals α. This shows that, as expected, the test is unbiased. As a numerical example, suppose that we ﬁx a signiﬁcance level α = 0.025 and we wish to test the simple hypotheses H0 : θ = 15.0; H1 : θ = 17.0 knowing that the standard deviation of the underlying normal population is

232 Mathematical statistics σ = 2.0. Suppose further that we carry out n = 20 measurements leading to m = 16.2. Since the rejection region in this case is '

2.0 1 = m ≥ 15.0 + 1.96 √ 20

( = {m ≥ 15.88}

and m = 16.2 falls in it, we reject the null hypothesis and accept H1 . Moreover, we can calculate the power of the test noting that r = −2.51 so that the corresponding upper quantile is 0.994 = W(θ1 ) and the probability of a type II error is β = 1 − 0.994 = 0.006. Following the same line of reasoning as above, it is easy to determine that the rejection region in the case θ1 < θ0 is ' ( σ 1 = x : m ≤ θ0 − zα √ n

(6.14b)

because we get the condition Pθ0 [l(X) ≤ c] = Pθ0 [Z ≤ t(c)] = α thus implying that now t(c) – where t(c) is as in eq. (6.13b) – is the α-lower quantile of the standard normal distribution. Owing to the symmetry of the distribution, this lower quantile is −zα and therefore eq. (6.14b) follows. As a further development of the exercise, consider the following problem: in the case θ1 > θ0 we have ﬁxed the probability of a type I error to a value α, what (minimum) sample size do we need to obtain a probability of type II error smaller than a given value β? The probability of a type II error is √ √ Pθ1 (M < θ0 + zα σ/ n) = Pθ1 {Z < zα − n(θ1 − θ0 )/σ } √ where Z is the standard normal√r.v. Z = n(M − θ1 )/σ . The desired upper limit β is obtained when zα − n(θ1 − θ0 )/σ equals the β-lower quantile of the standard normal distribution. If we denote this lower quantile by qβ we get n=

σ 2 (zα − qβ )2 (θ1 − θ0 )2

(6.15)

and consequently n˜ = [n] + 1 (the square brackets denote the integer part of the number) is the minimum required sample size. So, for instance, taking the same numerical values as above for α, θ0 , θ1 , σ , suppose we want a maximum probability of type II error β = 0.001. Then, the minimum sample size is n˜ = 26 (because z0.025 = 1.96, q0.001 = −3.09 and eq. (6.15) gives n = 25.5). Example 6.1(b) Consider now a normal population with known mean µ and unknown variance σ 2 = θ 2 . Somehow we know that the variance is

The test of statistical hypotheses θ02

θ12

θ12

either or (with of simple hypotheses

>

θ02 )

233

and the scope of the analysis is to test the pair

H0 : θ 2 = θ02

(6.16)

H1 : θ 2 = θ12

Again, we use Neyman–Pearson’s lemma to obtain the most powerful test for the case at hand. Since

n 1 1 1 2 (xi − µ) exp − − 2 l(x) = 2 θ02 θ1 i=1 n

n θ12 − θ02 θ1 xi − µ 2 = exp − θ0 θ0 2θ12 i=1

θ1 θ0

n

the inequality deﬁning the rejection region (6.9) holds if

n xi − µ 2 θ0

i=1

≥

2θ12 θ12

− θ02

[n log(θ1 /θ0 ) − log c] ≡ t(c)

(6.17)

Under the hypothesis H0 , each one of the n independent r.v.s Yi = [(Xi − µ)/θ0 ]2 is distributed according to the χ 2probability law with one degree of freedom. Consequently, the sum Y = Yi has a χ 2 distribution with n degrees of freedom and the relation Pθ0 [l(X) ≤ c] = Pθ0 [Y ≥ t(c)] = α means that t(c) must be the α-upper quantile of this distribution. Then, denoting 2 , the rejection region for the test is this quantile by the symbol χα;n 1 = x :

n

6 2

(xi − µ) ≥

2 θ02 χα;n

(6.18a)

i=1

As a numerical example, suppose we ﬁx a signiﬁcance level α = 0.05 and we wish to test the hypotheses H0 : θ 2 = 3.0; H1 : θ 2 = 3.7 for a normal population with mean µ = 18. If we carry out an experiment consisting of, say, 15 measurements x = (x1 , . . . , x15 ), our rejection region will be 1 = x :

15 i=1

6 2 (xi − 18)2 ≥ 3.0χα;n

= x:

15 i=1

6 (xi − 18)2 ≥ 74.988

234 Mathematical statistics 2 because from statistical tables we get the upper quantile χ0.05;15 = 24.996. 2 2 It is left to the reader to show that in the case θ1 < θ0 the rejection region is

1 = x :

n

6 2

(xi − µ) ≤

2 θ02 χ1−α;n

(6.18b)

i=1 2 where χ1−α;n is the (1 − α)-upper quantile (or, equivalently, the α-lower quantile) of the χ 2 distribution with n degrees of freedom. So, the basic idea of Proposition 6.1 is rather intuitive: since the likelihoodratio statistics (6.10) can be considered as a relative measure of the ‘weight’ of the two hypotheses , l(x) > 1 suggests that the observed data support the null hypothesis while the relation l(x) < 1 tends to imply the opposite conclusion and the speciﬁc value of l(x) – that is c in eq. (6.9) – below which we reject H0 depends on α, that is, the risk we are willing to take of making a type I error. In this light, therefore, it is evident that nothing would change if, as some authors do, one deﬁned the likelihood-ratio as l(X) = L(X; θ1 )/L(X; θ0 ) and considered the rejection region 1 = {l(x) ≥ c} with Pθ0 [l(X) ≥ c] = α. When the probability distributions are discrete the same line of reasoning leads to the most powerful test for the simple hypotheses (6.8). Discreteness, however, often introduces one minor inconvenience. In fact, since the likelihood-ratio statistic takes on only discrete values, say l1 , l2 , . . . , lk , . . ., it may not be possible to satisfy the condition Pθ0 [l(X) ≤ c] = α exactly. The following example will clarify this situation.

Example 6.2 At the signiﬁcance level α, suppose that we want to test the simple hypotheses (6.8) (with θ1 > θ0 ) on the unknown parameter p = θ of a binomial model. Deﬁning y = ni=1 xi we have l(x) =

θ0 θ1

y

1 − θ0 1 − θ1

n−y

and l(x) ≤ c if

1 − θ0 −1 1 − θ0 θ1 + log − log c ≡ t(c) n log y ≥ log θ0 1 − θ1 1 − θ1 Under the null hypothesis, the r.v. Y = X1 + · · · + Xn – being the sum of n binomial r.v.s – is itself binomially distributed with parameter θ0 , and in order to meet the condition Pθ0 [l(X) ≤ c] = Pθ0 [Y ≥ t(c)] = α exactly there should exist an (integer) index k = k(α) such that n n m θ (1 − θ0 )n−m = α m 0

m=k(α)

(6.19)

The test of statistical hypotheses

235

If such an index does exist – a rather rare occurrence indeed – the test attains the desired signiﬁcance level and the rejection region is 1 = x : y =

n

6 xi ≥ k(α)

(6.20)

i=1

However, the most common situation by far is the case in which eq. (6.19) is not satisﬁed exactly but we can ﬁnd an index r = r(α) such that n n m θ (1 − θ0 )n−m < α < m 0

α ≡

m=r(α)

n m=r(α)−1

n m θ (1 − θ0 )n−m ≡ α m 0 (6.21)

At this point we can deﬁne the rejection region as (a) 1 = {x : y ≥ r(α)} or as (b) 1 = {x : y ≥ r(α) − 1}, knowing that in both cases we do not attain the desired level α but we are reasonably close to it. In case (a), in fact, the actual signiﬁcance level α is slightly lower than α (r(α) is the minimum index satisfying the left-hand side inequality of (6.21)) while in case (b) the actual signiﬁcance level α is slightly greater than α (r(α)−1 is the maximum index satisfying the right-hand side inequality of (6.21)). Also, in terms of power we have n n m 1−β = θ (1 − θ1 )n−m < m 1

m=r(α)

n

m=r(α)−1

n m θ (1 − θ1 )n−m = 1 − β m 1

and, as expected, β > β . In the two cases, respectively, Proposition 6.1 guarantees that these are the most powerful tests at levels α and α . Besides the cases (a) and (b) – which in most applications will do – a third possibility called ‘randomization’ allows the experimenter to attain the desired level α exactly. Suppose that we choose the rejection region (b) associated to a level α > α and deﬁned in terms of the index s(α) ≡ r(α)−1. Under the null hypothesis, let us call P0 the probability of the event Y = s(α), that is, P0 ≡ Pθ0 {Y = s(α)} =

n θ s(α) (1 − θ0 )n−s(α) s(α) 0

(which, on the graph of the PDF F0 (x), is the jump F0 (r) − F0 (s)) and let us introduce the ‘critical (or rejection) function’ g(x) deﬁned as g(x) =

⎧ ⎨

1, y > s(α) (P0 + α − α )/P0 , y = s(α) ⎩ 0, y < s(α)

(6.22)

236 Mathematical statistics Then we reject H0 if y > s(α), we accept it if y < s(α) and, if y = s(α), we reject it with a probability (P0 + α − α )/P0 – or, equivalently, accept it with the complementary probability (α − α)/P0 . This means that if y = s(α) we have to set up another experiment with two possible outcomes, one with probability (P0 + α − α )/P0 and the other with probability (α − α)/P0 ; we reject H0 if the ﬁrst outcome turns out, otherwise we accept it. The probability of type I error of this randomized test (i.e. its signiﬁcance level) is obtained by taking the expectation of the critical function and we get, as expected P(H1 |H0 ) = Eθ0 [g(x)] = Pθ0 {y > s(α)} +

P0 + α − α Pθ0 {y = s(α)} P0

= α − P0 +

P0 + α − α P0 = α P0

The reader is invited to: (a) show that the case θ0 > θ1 leads to the rejection region 1 = {x : y ≤ r(α)} where, taking r(α) as the maximum index satisfying the inequality α ≡

r(α) n m=0

m

θ0m (1 − θ0 )n−m ≤ α

(6.23)

the attained signiﬁcance level α is slightly lower than α (unless we are so lucky to have the equal sign in (6.23)); (b) work out the details of randomization for this case. So, in the light of Example 6.2 we can make the following general considerations on discrete cases: (i) Carrying out a single experiment, discreteness generally precludes the possibility of attaining the speciﬁed signiﬁcance level α exactly. Nonetheless we can ﬁnd a most powerful test at a level α < α or at a level α > α. (ii) At this point, we can either be content of α (or α , whichever is our choice) or – if the experiment leads to a likelihood-ratio value on the border between the acceptance and rejection regions – we can ‘randomize’ the test in order to attain α. In the ﬁrst case we lack the probability α − α while we have a probability α − α in excess in the second case. Broadly speaking, randomization compensates for this part by adding a second experimental stage.

The test of statistical hypotheses

237

(iii) This second stage can generally be carried out by looking up a table of random numbers. So, referring to Example 6.2, suppose that we get (P0 + α − α )/P0 = 0.65 and before the experiment we have arbitrarily selected a certain position (say, 12th from the top) of a certain column at a certain page of a two-digit random numbers table. If that number lies between 00 and 64 we reject H0 and accept it otherwise. By so doing, we have performed the most powerful test of size α on the simple hypotheses (6.8). 6.3.2

A few notes on sequential analysis

So far we have considered the sample size n as a number ﬁxed in advance. Even at the end of Example 6.1(a), once we have chosen the desired values of α and β, eq. (6.15) shows that n can be determined before the experiment is carried out. A different approach due to Abraham Wald and called ‘sequential analysis’ leads to a decision on the null hypothesis without ﬁxing the sample size in advance but by considering it as a random variable which depends on the experiment’s outcomes. It should be pointed out that sequential analysis is a rather broad subject worthy of study in its own right (see, for instance, Wald’s book [22]) but here we limit ourselves to some general comments relevant to our present discussion. As in the preceding section, suppose that we wish to test the two simple hypotheses (6.8). For k = 0, 1 let Lkm ≡ L(x1 , . . . , xm ; θk ) =

m

fk (xi )

(6.24)

i=1

be the two likelihood functions L0m , L1m under the hypothesis H0 , H1 , respectively, after m observations (i.e. the realization x1 , . . . , xm ). Then, the general idea of Wald’s sequential test is as follows : (i) we appropriately ﬁx two positive numbers r, R (r < 1 < R), (ii) we continue testing as long as the likelihood ratio lm ≡ L0m /L1m lies between the two limits r, R and (iii) terminate the process for the ﬁrst index which violates one of the inequalities r

θ0 , (b) H1 : θ < θ0 or (c) H1 : θ = θ0 and one speaks of one-sided alternative in cases (a) and (b) – right- and left-sided, respectively – and of two-sided alternative in case (c). With composite hypotheses , a uniformly most powerful (ump) test exists only for some special classes of problems but many of these, fortunately, occur quite often in practice. So, for instance, many statistical models for which there is a sufﬁcient statistic T (for the parameter θ under test so that eq. (5.52) holds) have a monotone (in T) likelihood ratio; for these models it can be shown (see Ref. [10] or [15]) that a ump test to verify H0 : θ = θ0 against a one-sided alternative does exist. This ‘optimal’ test, moreover, coincides with the Neyman–Pearson’s test for H0 : θ = θ0 against an arbitrarily ﬁxed alternative H1 , where H1 is in the form (a) or (b). Even more, the ﬁrst test is also the ump test for the doubly composite case H0 : θ ≤ θ0 ; H1 : θ > θ0 while the second is the ump test for H0 : θ ≥ θ0 ; H1 : θ < θ0 . In spite of all these interesting and important results , it is not our intention to enter into such details and we refer the interested reader to more specialized literature. Here, after some examples of composite hypotheses cases , we will limit ourselves to the description of the general method called ‘likelihood ratio test’ which – although not leading to ump tests in most cases – has a number of other desirable properties.

240 Mathematical statistics Example 6.3(a) If we wish to test the hypotheses H0 : θ = θ0 ; H1 : θ > θ0 for the mean of a normal model with known variance σ 2 , we can follow the same line of reasoning of Example 6.1(a) and obtain the rejection region (6.14a). Since this rejection region does not depend on the speciﬁc value θ1 against which we compare H0 (provided that θ1 > θ0 ), it turns out that this is the uniformly most powerful test for the case under investigation and, as noted above, for the pair of hypotheses H0 : θ ≤ θ0 ; H1 : θ > θ0 as well. Similar considerations apply to the problem H0 : θ = θ0 ; H1 : θ < θ0 and we can conclude that the rejection region (6.14b) provides the ump test for this case and for H0 : θ ≥ θ0 ; H1 : θ < θ0 . Example 6.3(b) Considering the normal model of Example 6.1(b) – that is, known mean and unknown variance σ 2 = θ 2 – it is now evident that the rejection region (6.18a) provides the ump test for the problem H0 : θ 2 ≤ θ02 ; H1 : θ 2 > θ02 while (6.18b) applies to the case H0 : θ 2 ≥ θ02 ; H1 : θ 2 < θ02 . In all the cases above, the probability of a type II error β (and therefore the power) will depend on the speciﬁc value of the alternative. Often, in fact, one can ﬁnd graphs of β plotted against an appropriate variable with the sample size n as a parameter. These graphs are called operating characteristic curves (OC curves ) and the variable on the abscissa axis depends on the type of test. So, for instance, the OC curve for the ﬁrst test of Example 6.3(a) plots β versus (θ1 − θ0 )/σ for some values of n. Fig. 6.1 is one such graph for α = 0.05 and the three values of sample size n = 5, n = 10 and n = 15. As it should be expected, β decreases as the difference θ1 − θ0 increases and, for a ﬁxed value of this quantity, β is lower for larger sample sizes. Similar curves can generally be drawn with little effort for the desired sample size by using widely available software packages such as, for instance,

Beta (prob. of type II error)

1.2 n=5 n =10 n =15

1.0 0.8 0.6 0.4 0.2 0.0 0.00

0.40

0.80

1.20 (1 – 0)/

1.60

Figure 6.1 One-sided test (size = 0.05 − H1 : θ > θ0 ).

2.00

2.40

The test of statistical hypotheses Excel® ,

Matlab®

241

etc. The reader is invited to do so for the cases of

Example 6.3(b). Example 6.3(c) Referring back to Example 6.3(a) – normal model with known variance – let us make some considerations on the two-sided case H0 : θ = θ0 ; H1 : θ = θ0 . At the signiﬁcance level α, the rejection regions for the one-sided alternatives H1 : θ > θ0 and H1 : θ < θ0 are given by eqs (6.14a) and (6.14b), respectively. If we conveniently rewrite these equations as √ n(m − θ0 )/σ ≥ zα } √ − 1 = {x : n(m − θ0 )/σ ≤ −zα } + 1 = {x :

(6.29)

we may think of specifying the rejection region 1 of the two-sided test (at + the level α) as 1,α = − 1,a ∪ 1,b , where a, b are two numbers such that a + b = α. Moreover, intuition suggests to take a ‘symmetric’ region by choosing a = b = α/2 thus obtaining √ ' ( n ˜ 1 = x : |m − θ0 | ≥ zα/2 σ

(6.30)

which, in other words , means that we reject the null hypothesis when θ0 is sufﬁciently far – on one side or the other – from the sample mean m. As before, the term ‘sufﬁciently far’ depends on the risk involved in rejecting a true null hypothesis (or, equivalently, accepting a false alternative). We do not do it here but these heuristic considerations leading to (6.30) can be justiﬁed on a more rigorous basis showing that, for the case at hand, eq. (6.30) is a good choice because it deﬁnes the ump test among the class of unbiased tests. In fact, it turns out that a ump test does not exist for this case because (at the level α) the two tests leading to (6.29) can be considered in their own right as tests for the alternative H1 : θ = θ0 . In this light, we already know that (i) the + 1 -test is the most powerful in the region θ > θ0 (ii) the − -test is the most powerful for θ < θ0 and (iii) both their powers take on 1 the value α at θ = θ0 . However, as tests against H1 : θ = θ0 , they are biased. In fact, the power W + (θ ) of the ﬁrst test is rather poor (i.e. low and such that W + (θ) < α) for θ < θ0 and the same holds true for W − (θ ) when θ > θ0 ˜ ) the power of the test (6.30), we have the inequalities so that, calling W(θ ˜ ) < W − (θ ), W + (θ) < α < W(θ

θ < θ0

˜ ) < W + (θ ), W − (θ) < α < W(θ

θ > θ0

˜ 0 ) = α. The fact that for two-sided while, clearly, W − (θ0 ) = W + (θ0 ) = W(θ alternative hypothesis there is no ump test but there sometimes exists a ump

242 Mathematical statistics unbiased test is more general than the special case considered here and the interested reader can refer, for instance, to [15] for more details. As pointed out above, the likelihood ratio method is a rather general procedure used to test composite hypotheses of the type (6.2). In general, it does not lead to ump tests but gives satisfactory results in many practical problems. As before, let X = (X1 , . . . , Xn ) be a random sample and let the (absolutely continuous) model be expressed in terms of the pdf f (x). The likelihood ratio statistic is deﬁned as λ(X) =

supθ∈0 L(X; θ ) supθ∈ L(X; θ )

=

L(θˆ0 ) L(θˆ )

(6.31)

where θˆ0 is the maximum likelihood (ML) estimate of the unknown parameter θ when θ ∈ 0 and θˆ is the ML estimate of θ over the entire parameter space . Deﬁnition 6.31 shows that 0 ≤ λ ≤ 1 because we expect λ to be close to zero when the null hypothesis is false and close to unity when H0 is true. In this light, the rejection region is deﬁned by 1 = {x : λ(x) ≤ c}

(6.32)

where the number c is determined by the signiﬁcance level α and it is such that sup Pθ {λ(X) ≤ c} = α

θ∈0

(6.33)

which amounts to the condition P(H1 |H0 ) = Pθ {λ(X) ≤ c} ≤ α for all θ ∈ 0 (recall the deﬁnition of power (6.4) and eq. (6.5)). It is clear at this point that the likelihood ratio test generalizes Neyman–Pearson’s procedure to the case of composite hypotheses and reduces to it when both hypotheses are simple. If the null hypothesis is simple – that is, of the form H0 : θ = θ0 – the numerator of the likelihood ratio is simply L(θ0 ) and, since 0 contains only the single element θ0 , no ‘sup’ appears both at the numerator of (6.31) and in eq. (6.33). Example 6.4(a) In Example 6.3(a) we have already discussed the normal model with known variance when the test on the mean µ = θ is of the form H0 : θ ≤ θ0 ; H1 : θ > θ0 . Let us now examine it in the light of the likelihood ratio method. For this case, clearly, = R, 0 = (−∞, θ0 ] and 1 = (θ0 , ∞). The denominator of (6.31) is L(M) because the sample mean M is the ML estimator of the mean (recall Section 5.5) over the entire parameter space. On the other hand, it is not difﬁcult to see that the numerator of (6.31) is L(θ0 ) so that n λ(X) = exp − 2 (M − θ0 )2 2σ

(6.34)

The test of statistical hypotheses

243

and the inequality in (6.32) is satisﬁed if : |m − θ0 | ≥

2σ 2 log(1/c) = t(c) n

where, as usual, m is the realization of M. Since m > θ0 in the rejection region, the condition (6.33) reads √ '√ ( n(M − θ ) n(θ0 − θ + t(c)) ≥ σ σ θ≤θ0 √

nt(c) =P Z≥ =α σ

sup P{M ≥ θ0 + t(c)} = sup P

θ≤θ0

√ because in 1 the r.v. Z = n(M − θ )/σ is standard normal and the ‘sup’ of the probability above is attained when θ = θ0 . The conclusion is that, as expected, the rejection region is given by eq. (6.14), that is, the same as for the simple case H0 : θ = θ0 ; H1 : θ = θ1 with θ1 > θ0 . Also, from the considerations above we know that eq. (6.14a) deﬁnes the ump for the problem at hand. Example 6.4(b) The reader is invited to work out the details of the likelihood ratio method for the normal case in which the hypotheses on the mean are H0 : θ = θ0 ; H1 : θ = θ0 and the variance is known. We have in this case = R, 0 = {θ0 } and 1 = (−∞, θ0 ) ∪ (θ0 , ∞) and, as above, λ(X) = L(θ0 )/L(M) so that eq. (6.34) still applies. Now, however, no √‘sup’ needs to be taken in eq. (6.33) which, in turn, becomes P{|Z| ≥ n t(c)/σ } = α thus leading to the rejection region (6.30) where, we recall, zα/2 denotes the α/2-upper quantile of the standard normal distribution. So, for instance, if α = 0.01 we have zα/2 = z0.005 = 2.576 while for a 10 times smaller probability of type I error – that is, α = 0.001 – we ﬁnd in tables the value zα/2 = z0.0005 = 3.291. The fact that (a) testing hypotheses on the mean of a normal model with known variance leads to rejection regions where the quantiles of the standard normal distribution appear; (b) testing hypotheses on the variance (Examples 6.1(b) and 6.3(b)) of normal model with known mean brings into play the quantiles of the χ 2 distribution with n degrees of freedom may suggest a connection between parametric hypothesis testing and interval estimation problems (Section 5.6). The connection, in fact, does exist and is

244 Mathematical statistics rather strong. Before considering the situation from a general point of view, we give two more examples that conﬁrm this state of affairs. Example 6.5(a) Consider a normal model with the same hypotheses on the mean as in Example 6.4(b) but with the important difference that the variance σ 2 is now unknown (note that in this situation the null hypothesis H0 : θ = θ0 is not simple because it does not uniquely determine the probability distribution). In this case, the parameter space = R × R+ is divided into the two sets 0 = {θ0 } × R+ and 1 = [(−∞, θ0 ) ∪ (θ0 , ∞)] × R+ . Under the null hypothesis θ = θ0 the maximum of the likelihood function is attained when σ 2 = σ02 = n−1 i (xi − θ0 )2 and the numerator of (6.31) is L(θ0 , σ02 ) = (2π σ02 e)−n/2 2 2 −n/2 Similarly, the denominator of (6.31) 2is given by L(M, S ) = (2π S e) 2 −1 because M and S = n i (Xi − M) , respectively, are the ML estimators of the mean and the variance. Consequently, if s2 is the realization of the sample variance S2 , we get

λ=

σ02 s2

−n/2

t2 = 1+ n−1

−n/2 (6.35)

√ where we set t = t(x) = n − 1(m − θ0 )/s and the second relation is easily obtained by noting that σ02 = s2 + (m − θ0 )2 . Equation (6.35) shows that there is a one-to-one correspondence between λ and t 2 and therefore the inequality in (6.32) is equivalent to |t| ≥ c (where c is as appropriate). Since the r.v. t(X) is distributed according to a Student distribution with n−1 degrees of freedom, the boundary c of the rejection region (at the signiﬁcance level α) must be the α/2-upper quantile of this distribution. Denoting this upper quantile by tα/2;n−1 we have 6 √ n−1 |m − θ0 | ≥ tα/2;n−1 1 = x : s

(6.36)

Alternatively, if one prefers to do so, 1 can be speciﬁed in terms of the ¯2 (1 − α/2)-lower quantile √ and/or using √ the unbiased estimator S instead of 2 S (and recalling that n − 1/s = n/¯s). As a numerical example, suppose we carried out n = 10 trials to test the hypotheses H0 : θ = 35; H1 : θ = 35 at the level α = 0.01. The rejection region is then 1 = {x : (3/s)|m − 35| ≥ 3.25} because the 0.005-upper quantile of the Student distribution with 9 degrees of freedom is t0.005; 9 = 3.250. So, if the experiment gives, for instance, m = 32.6 and s = 3.3 we fall outside 1 and we accept the null hypothesis at the speciﬁed signiﬁcance level.

The test of statistical hypotheses

245

In the lights of the comments above, the conclusion is as expected: testing the mean of a normal model with unknown variance leads to the appearance of the Student quantiles, in agreement with Example 5.9(b) where, for the same model, we determined a conﬁdence interval for the mean (eq. (5.76)). Example 6.5(b) At this point it should not be surprising if we say that testing the variance of a normal model with unknown mean involves the quantiles of the χ 2 distribution with n − 1 degrees of freedom (recall Example 5.10(b)). It is left to the reader to use the likelihood ratio method to determine that the rejection region for the hypotheses H0 : θ 2 = θ02 ; H1 : θ 2 = θ02 on the variance is 6 , , s¯2 s¯2 2 2 1 = x : 0 ≤ n − 1 2 ≤ χ1−α/2;n−1 ∪ n − 1 2 ≥ χα/2;n−1 θ0 θ0 (6.37)

2 2 , respectively, are the (1−α/2)-upper quantile and χα/2;n−1 where χ1−α/2;n−1 and the α/2-upper quantile of the χ 2 distribution with n − 1 degrees of freedom. So, for instance, if n = 10 and we are testing at the level α = 0.05 2 2 2 2 we ﬁnd χ1−α/2;n−1 = χ0.975;9 = 2.70 and χα/2;n−1 = χ0.025;9 = 19.023.

Therefore, we will accept the null hypothesis H0 if 2.70 < 3¯s2 /θ02 < 19.023. 6.3.4

Complements on parametric hypothesis testing

In addition to the main ideas of parametric hypotheses testing discussed in the preceding sections, some further developments deserve consideration. Without claim of completeness, we do this here by starting from where we left off in Section 6.3.3: the relationship with conﬁdence intervals estimation. 6.3.4.1

Parametric tests and conﬁdence intervals

Let us ﬁx a signiﬁcance level α. If we are testing H0 : θ = θ0 against the composite alternative H1 : θ = θ0 , the resulting rejection region 1 will depend, for obvious reasons , on the value θ0 and so will the acceptance region 0 = C 1 . As θ0 varies in the parameter space we can deﬁne the family of sets 0 (θ ) ⊂ , where each 0 (θ ) corresponds to a value of θ. On the other hand, a given realization of the sample x will fall – or will not fall – in the acceptance region depending on the value of θ under test and those values of θ such that x does fall in the acceptance region deﬁne a subset G ⊂ of the parameter space. By letting x free to vary in , therefore, we can deﬁne the family of subsets G(x) = {θ : x ∈ 0 (θ )} ⊂ . The consequence of this (rather intricate at ﬁrst sight) construction of subsets in the sample and parameter space is the fact that the events {X ∈ H0 (θ )} and

246 Mathematical statistics {θ ∈ G(X)} are equivalent, and since the probability of the ﬁrst event is 1 − α so is the probability of the second. Recalling Section 5.6, however, we know that this second event deﬁnes a conﬁdence interval for the parameter θ. By construction, the conﬁdence level associated to this interval is 1 − α = γ . The conclusion, as anticipated, is that parametric hypothesis testing and conﬁdence interval estimation are two strictly related problems and the solution of one leads immediately to the solution of the other. Moreover, a ump test – if it exists – corresponds to the shortest conﬁdence interval and vice versa. Having established the nature of the connection between the two problems, we can now reconsider some of the preceding results from this point of view. Let us go back to Example 6.3(c) where the rejection region is given by eq. (6.30). This relation implies that the ‘acceptance’ subsets 0 (θ ) have the form ( ' σ σ 0 (θ) = x : m − √ zα/2 < θ < m + √ zα/2 n n

(6.38)

Those x such that x ∈ 0 (θ ) correspond to the values of θ which satisfy the inequality in (6.38) and these θ, in turn, deﬁne the corresponding set √ G(x). From the discussion above it follows that (M ± zα/2 σ/ n) is a 1 − α conﬁdence interval for the mean of a normal model with known variance. This is in agreement with eq. (5.75) of Example 5.9(a) because the α/2-upper quantile zα/2 (of the standard normal distribution) is the (1 − α/2)-lower quantile and, since 1 − α = γ , this is just the (1 + γ )/2-lower quantile (which was denoted by the symbol c(1+γ )/2 in Example 5.9(a)). Similarly, for the same statistical model we have seen that the rejection region to test H0 : θ = θ0 ; H1 : θ > θ0 is given by√(6.14a). This implies that the acceptance sets are 0 (θ ) = {x : m < θ +zα σ/√ n} and the corresponding sets G(x) are given by G(x) = {θ : θ > m − zα σ/ n}. The conclusion is that

σ M − zα √ , +∞ n

(6.39a)

is a lower (1 − α)-conﬁdence (one-sided) interval for the mean. The interval (6.39a) is called ‘lower’ because only the lower limit for θ (recall eq. (5.69a)) is speciﬁed; by the same token, it is evident that we can use the rejection region (6.14b) to construct the upper (1 − α)-conﬁdence interval

σ −∞, M + zα √ n

(6.39b)

The argument, of course, works in both directions and we can, for instance, start from the γ -CI (5.76) for the mean of a normal model with unknown

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247

variance to write the acceptance region for the test H0 : θ = θ0 ; H1 : θ = θ0 as √ ' ( n(m − θ0 ) 0 = x : −t(1+γ )/2;n−1 < < t(1+γ )/2;n−1 s¯ from which we get the rejection region 1 = C 0 of eq. (6.36) by noting that √ √ (i) n − 1/s = n/¯s and (ii) (1 + γ )/2 = 1 − α/2 so that the Student lower quantiles of (5.76) are the 1 − (1 − α/2) = α/2 (Student) upper quantiles of eq. (6.36). At this point it is left to the reader to work out the details of the parametric hypothesis counterparts of Examples 5.11 (a–c) by determining the relevant acceptance and rejection regions. 6.3.4.2

Asymptotic behaviour of parametric tests

The second aspect we consider is the asymptotic behaviour of parametric tests. As a ﬁrst observation, we recall from Chapter 5 that a number of important sample characteristics are asymptotically normal with means and variances determined by certain population parameters. Consequently, when the test concerns one of these characteristics the ﬁrst thing that comes to mind is, for large samples, to use the normal approximation by replacing any unknown population parameter by its (known) sample counterpart thus obtaining a rejection region determined by the appropriate quantile of the standard normal distribution. This is a legitimate procedure but it should be kept in mind that it involves two types of approximations (i) the normal approximation for the distribution of the characteristic under test and (ii) the use of sample values for the relevant unknown population parameters. So, in practice, it is often rather difﬁcult to know whether our sample is large enough and our test has given a reliable result. As a rule of thumb, n > 30 is generally good enough when we are dealing with means while n > 100 is advisable for variances, medians , coefﬁcients of skewness and kurtosis. For some other ‘less tractable’ characteristics, however, even samples as large as 300 or more do not always give a satisfactory approximation. Let us now turn our attention to Neyman–Pearson’s lemma on simple hypotheses (Proposition 6.1). The boundary value c in eq. (6.9) can only be calculated when we know the distribution of the statistic l(X) under H0 (and, similarly, the probability β of a type II error can be calculated when we know the distribution of l(X) under H1 ). Since this is not always possible, we can proceed as follows. If we deﬁne the r.v.s Yi = log

f0 (Xi ) f1 (Xi )

(6.40)

for i = 1, 2, . . . , n then Sn = Y1 + Y2 + · · · + Yn is the sum of n iid variables. Depending on which hypothesis is true, its mean and variance are

248 Mathematical statistics Eθ0 (Sn ) = na0 and Varθ0 (Sn ) = nσ02 under H0 or Eθ1 (Sn ) = na1 and Varθ1 (Sn ) = nσ12 under H1 , where we called a0 , σ02 and a1 , σ12 the mean and variance (provided that they exist) of the variables Yi under H0 and H1 , respectively. At this point, the CLT √ (Proposition 4.22) tells us that, under H0 , the r.v. Z = (Sn − na0 )/σ0 n is asymptotically standard normal and, since deﬁnition (6.40) implies that the inequality in eq. (6.9) is equivalent to Sn ≤ log c, for sufﬁciently large values of n we can write ' Pθ0

log c − na0 Z≤ √ σ0 n

( =α

(6.41)

which, on the practical side, implies that we have an approximate rejection region: we reject the null hypothesis if the realization of the sample is such that (y1 + y2 + · · · + yn ) − na0 ≤ cα √ σ0 n

(6.42)

where cα is the α-lower quantile of the standard normal distribution. Clearly, the goodness of the approximation depends on how fast the variable Z converges (in distribution) to the standard normal r.v.; if the rate of convergence is slow, a rather large sample is required to obtain a reliable test. In regard to the more general likelihood ratio method, it is convenient to consider the monotone function (X) = −2 log λ(X) of the likelihood ratio λ(X). Provided that the regularity conditions for the existence, uniqueness and asymptotic normality of the ML estimate θˆ of the parameter θ are met (see Sections 5.5 and 5.5.1), it can be shown that the asymptotic rejection region for testing the null hypothesis H0 : θ = θ0 is given by 2 1 = {x : (x) ≥ χ1−α;1 }

(6.43)

2 where χ1−α;1 is the (1 − α)-lower quantile of the χ 2 distribution with one degree of freedom. We do not prove this assertion here but we note that eq. (6.43) is essentially due to the fact that, under H0 , we have

(i) (X) → χ 2 (1)[D]; 2 (ii) Pθ0 {(X) ≥ χ1−α; 1} → α as n → ∞. The result can also be extended directly to the case of a vector, say k-dimensional, parameter q = (θ1 , . . . , θk ) and it turns out that the rejection region is deﬁned by means of the (1−α)-lower quantile of the χ 2 distribution with k degrees of freedom. In addition, the procedure still applies if the null hypothesis speciﬁes only a certain number, say r, of the k components of q. In this case the numerator of (6.31) is obtained by maximizing L with

The test of statistical hypotheses

249

respect to the remaining k − r components and r is the number of degrees of freedom of the asymptotic distribution. Also, another application of the result stated by eq. (6.43) is the construction of conﬁdence intervals for a general parametric model. In fact, eq. (6.43) implies that the (asymptotic) acceptance region is 0 = {x : (x) < χγ2;1 }, where γ = 1 − α. In the light of the relation between hypothesis testing and CIs we have that G(X) = {θ : (X) < χγ2;1 } is an asymptotic γ -CI for the parameter θ because Pθ {θ ∈ G(X)} → γ as n → ∞. In practice G(X) – called a maximum likelihood conﬁdence interval – can be used as an approximate CI when the sample size is large. All these further developments , however – together with the proof of the theorem above – are beyond our scope. For more details the interested reader may refer, for instance, to [1, 2, 10, 13, 19]. 6.3.4.3

The p-value: signiﬁcance testing

A third aspect worthy of mention concerns a slightly different implementation of the hypothesis testing procedure shown in the preceding sections. This modiﬁed procedure is often called ‘signiﬁcance testing’ and is becoming more and more popular because it somehow overcomes the rigidity of hypothesis testing. The main idea of signiﬁcance testing originates from the fact that the choice of the level α is, to a certain extent, arbitrary and the common values 0.05, 0.025 and 0.01 are often used out of habit rather than through careful analysis of the consequences of a type I error. So, instead of ﬁxing α in advance we perform the experiment, calculate the value of the appropriate test statistic and report the so-called p-value (or ‘observed signiﬁcance level’ and denoted by αobs ), deﬁned as the smallest value of α for which we reject the null hypothesis. Let us consider an example. Example 6.6 Suppose that we are testing H0 : θ = 100 against H1 : θ < 100 where the parameter θ is the mean of a normally distributed r.v. with known variance σ 2 = 25. Suppose further that an experiment on a sample of n = 16 products gives the sample mean m = 97.5. Since the rejection region for this case is given by eq. (6.14b), a test at level α = 0.05 (z0.05 = 1.645) leads to reject H0 in favour of H1 and the same happens at the level α = 0.025 (z0.025 = 1.960). If, however, we choose α = 0.01 (z0.01 = 2.326) the conclusion is that we must accept H0 . This kind of situation is illustrative of the rigidity of the method; we are saying in practice that we tolerate 1 chance in 100 of making a type I error but at the same time we state that 2.5 chances in 100 is too risky. One way around this problem is, as noted above, to calculate the p-value and √ move on from there. For the case at hand the relevant test statistic n(M − θ0 )/σ is standard normal and attains the value 4(−2.5)/5 = −2.0 which, we ﬁnd on statistical tables, corresponds to the level αobs = 0.0228.

250 Mathematical statistics This is the p-value for our case, that is, the value of α that will just barely cause H0 to be rejected. In other words, on the basis of the observed data we would reject the null hypothesis for any level α ≥ αobs = 0.0228 and accept it otherwise. Reporting the p-value, therefore, provides the necessary information to the reader to decide whether to accept or reject H0 by comparing this value with his/her own choice of α: if one is satisﬁed with a level α = 0.05 then he/she will not accept H0 but if he/she thinks that α = 0.01 is more appropriate for the case at hand, the conclusion is that H0 cannot be rejected. (As an incidental remark, it should be noted that the calculation of the p-value for this example is rather easy but it may not be so if the test statistic is not standard normal and we must rely only on statistical tables. However, the use widely available software packages has made things much easier because p-values are generally given by the software together with all the other relevant results of the test.) The example clariﬁes the general idea. Basically, this approach provides the desired ﬂexibility; if an experiment results in a low p-value, say αobs < 0.01, then we can be rather conﬁdent in our decision of rejecting the null hypothesis because – had we tested it at the ‘usual’ levels 0.05, 0.025 or 0.01 – we would have rejected it anyway. Similarly, if αobs > 0.1 any one of the usual testing levels would have led to the acceptance of H0 and we can feel quite comfortable with the decision of accepting H0 . A kind of ‘grey area’, so to speak, is when 0.01 < αobs < 0.1 and, as a rule of thumb, we may reject H0 for 0.01 < αobs < 0.05 and accept it for 0.05 < αobs < 0.1. It goes without saying, however, that exceptions to this rule are not rare and the speciﬁc case under study may suggest a different choice. In addition, we must not forget to always keep an eye on the probability β of a type II error. 6.3.4.4

Closing remarks

To close this section, a comment of general nature is not out of place: in some cases , taking a too large sample size may be as bad an error as taking a too small sample size. The reason lies in the fact as n increases we are able to detect smaller and smaller differences from the null hypothesis (this, in other words, means that that the ‘discriminating power’ of the test increases as n increases) and consequently we will almost always reject H0 if n is large enough. In performing a test, therefore, we must keep in mind the difference between statistical signiﬁcance and practical signiﬁcance, where this latter term refers to both reasonableness and to the nominal speciﬁcations , if any, for the case under study. So, without loss of generality, consider the test of Example 6.6 and suppose that for all practical purposes it would not matter much if the mean of the population were within ± 0.5 units from the value θ0 = 100 under test. If we decided to take a sample of n = 1600 products obtaining, say, the sample mean m = 99.7, we would reject

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251

the null hypothesis at α = 0.05, α = 0.025 and α = 0.01 (incidentally, the p-value in this case is αobs = 0.0082). This is clearly unreasonable because the statistically signiﬁcant difference detected by the test (which leads to the rejection of H0 ) does not correspond to a practical difference and we put ourselves in the same ironical situation of somebody who uses a microscope when a simple magnifying glass would do. As stated at the beginning of this section, these complementary notes do not exhaust the broad subject of parametric hypothesis testing. In particular, the various methods classiﬁed under the name ‘analysis of variance’ (generally denoted by the acronym ANOVA) have not been considered. We just mention here that the simplest case of ANOVA consists in comparing the unknown means µ1 , µ2 , . . . , µk of a number k > 2 of normal populations by testing the null hypothesis H0 : µ1 = µ2 = · · · = µk against the alternative that at least one of the equalities does not hold. As we can immediately see, this is an important case of parametric test where – it can easily be shown – pair wise comparison is not appropriate. For a detailed discussion of this interesting topic and of its ramiﬁcations the reader may refer to [6, 7, 9, 12, 18].

6.4

Testing the type of distribution (goodness-of-ﬁt tests)

In the preceding sections we concerned ourselves with tests pertaining to one (or more) unknown parameter(s) of a known distribution law, thus tacitly implying that we already have enough evidence on the underlying probability distribution – that is, normal, Poisson, binomial or other – with which we are dealing. Often, however, the uncertainty is on the type of distribution itself and we would like to give more support to our belief that the sample X = (X1 , . . . , Xn ) is, in fact, a sample from a certain distribution law. In other words, this means that on the basis of n independent observations of a r.v. X with unknown distribution FX (x) we would like to test the null hypothesis H0 : FX (x) = F(x) against H1 : FX (x) = F(x), where F(x) is a speciﬁed probability distribution. Two of the most popular tests for this purpose are Pearson’s χ 2 -test and Kolmogorov–Smirnov test. 6.4.1

Pearson’s χ 2 -test and the modiﬁed χ 2 -test

Let us start with Pearson’s test by assuming at ﬁrst that F(x) is completely speciﬁed – that is, it is not of the form F(x; θ ), where θ is some unknown parameter of the distribution. Now, let D1 , D2 , . . . , Dr be a ﬁnite partition of the space D of possible values of X (i.e. D = ∪rj=1 Dj and Di ∩ Di = ∅ for i = j) and let pj = P(X ∈ Dj |H0 ) be the probabilities of the event {X ∈ Dj } under H0 . These probabilities – which can be arranged in a r-dimensional vector p = (p1 , . . . , pr ) – are known because they depend on F(x) which,

252 Mathematical statistics in turn, is assumed to be completely speciﬁed. In fact, for j = 1, . . . , r we have P(X = xk ) (6.44a) pj = k : xk ∈Dj

if X is discrete and pj = f (x) dx

(6.44b)

Dj

if X is absolutely continuous with pdf f (x) = F (x). Turning now our attention to the sample X = (X1 , . . . , Xn ), let Nj be the r.v. representing the number of its elements falling in Dj so that νj = Nj /n is a r.v. representing the relative frequency of occurrence pertaining to Dj . Clearly, N1 + · · · + Nr = n or, equivalently r

νj = 1

(6.45)

j=1

With these deﬁnitions the hypotheses under test can be re-expressed as H0 : pj = νj and H1 : pj = νj for j = 1, . . . , r, thus implying that we should accept H0 when the sample frequencies Nj are in reasonable agreement with the ‘theoretical’ (assumed) frequencies npj . It was shown by Pearson that an appropriate test statistic for this purpose is T≡

r (Nj − npj )2 j=1

npj

=

r N2 j j=1

npj

−n

(6.46)

for which, under H0 , we have E(T) = r − 1

⎞ ⎛ r 1 ⎝ 1 − r2 − 2r + 2⎠ Var(T) = 2(r − 1) + n pj

(6.47)

j=1

and, most important, as n → ∞ T → χ 2 (r − 1) [D]

(6.48)

Equation (6.48) leads directly to the formulation of Pearson’s χ 2 goodnessof-ﬁt test: at the signiﬁcance level α, the approximate rejection region to test

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253

the null hypothesis H0 : pj = νj is given by 2 1 = {x : t(x) ≥ χ α;r−1 }

(6.49)

2 is the α-upper where t(x) is the sample realization of the statistic T and χ α;r−1 2 quantile of the χ distribution with r − 1 degrees of freedom. Although we do not prove the results (6.47) and (6.48) – the interested reader can refer to [4] or [10] – a few comments are in order:

(i) Explicitly, t(x) = j (nj − npj )2 /npj where nj (j = 1, . . . , r) is the realization of the r.v. Nj ; in other words, once we have carried out the experiment leading to the realization of the sample x = (x1 , . . . , xn ), n1 is the number of elements of x whose values fall in D1 , n2 is the number of elements of x whose values fall in D2 , and so on. (ii) The rejection region (6.49) is approximate because eq. (6.48) is an asymptotic relation; similarly, the equation P(T ∈ 1 |H0 ) = α deﬁning the probability of a type I error is strictly valid only in the limit of n → ∞. However, the quality of the approximation is generally rather good for n ≥ 50. For better results, many authors recommend to choose the Dj so that npj ≥ 5 for all j, or, at least, for more than 80–85% of all j (since we divide by npj in calculating T, we do not want the terms with the smaller denominators to ‘dominate’ the sum(6.46)). (iii) On the practical side, the choice of the Dj – which, for a onedimensional random variable, are non-overlapping intervals of the real line – plays an important role. Broadly speaking, they should not be too few and they should not be too many, that is, r should be neither too small nor too large. A possible suggestion (although in no way a strict rule) is to use the formula 0.2 2(n − 1)2 ∼ r=2 zα2

(6.50)

where zα is the α-upper quantile of the standard normal distribution. So, for the most common levels of signiﬁcance 0.05, 0.025 and 0.01 we have r ∼ = 1.883(n − 1)0.4 , r ∼ = 1.755(n − 1)0.4 and r ∼ = 1.639(n − 1)0.4 , respectively. Moreover, in order to comply with the indicative rule of point (ii) – that is, npj ≥ 5 for almost all j – adjacent end intervals (which cover the tails of the assumed distribution) are sometimes regrouped to ensure that the minimum absolute frequency is 5 or, at least, not much smaller than 5. Regarding the width of the intervals, it is common practice to choose equal-width intervals, although this requirement is not necessary. Some

254 Mathematical statistics authors, in fact, prefer to select the intervals so that the expected frequencies will be the same in each interval; excluding a uniform distribution hypothesis, it is clear that this choice will result in different interval widths. Example 6.7(a) Suppose that in n = 4000 independent trials the events A1 , A2 , A3 have been obtained n1 = 1905, n2 = 1015 and n3 = 1080 times, respectively. At the level α = 0.05 we want to test the null hypothesis H0 : p1 = 0.5; p2 = p3 = 0.25, where pj = P(Aj ). In this intentionally simple example the assumed distribution is discrete and np1 = 2000, np2 = np3 = 1000. Since the sample realization of the statistic 2 T is t = 11.14 and it is higher than the 0.05-upper quantile χ 0.05;2 = 5.99 we fall in the rejection region (6.49) and therefore, at the level α = 0.05, we reject the null hypothesis. Example 6.7(b) In n = 12000 tosses of a coin Pearson obtained n1 = 6019 heads and n2 = 5981 tails. Let us check at the levels 0.05 and 0.01 if this result is consistent with the hypothesis that he was using a fair coin (i.e. H0 : p1 = p2 = 0.5). We get now t = 0.120 and this value is lower than 2 2 both upper quantiles χ 0.05;1 = 3.841 and χ 0.01;1 = 6.635. Since we fall in the acceptance region in both cases, we accept H0 and infer that the data agree with the hypothesis of unbiased coin. The study of the power W of Pearson’s χ 2 test when H0 is not true is rather involved, also in the light of the fact that W cannot be computed unless a speciﬁed alternative H1 is considered. Nonetheless, an important result is worthy of mention: for every ﬁxed set of probabilities p¯ = p the power ¯ tends to unity as n → ∞ [10]. This, in words, is expressed function W(p) by saying that the test is ‘consistent’ (not to be confused with consistency for an estimator introduced in Section 5.4) and means that, under any ﬁxed alternative H1 , the probability 1 − β of rejecting the null hypothesis when it is false tends to 1 as n increases. It is evident that consistency, for any statistical hypothesis test in general, is a highly desirable property. A modiﬁed version of the χ 2 goodness-of-ﬁt test can be used even when the assumed probability distribution F(x) contains some unknown parameters, that is, it is of the form F(x; q) where q = (θ1 , . . . , θk ) is a set of k unknown parameters. In this case the null hypothesis is clearly composite – in fact, it identiﬁes a class of distributions and not one speciﬁc distribution in particular – and the statistic T itself will depend on q through the probabilities pj , that is, T = T(q) where T(q) =

r [Nj − npj (q)]2 j=1

npj (q)

=

r Nj2 j=1

npj (q)

−n

(6.51)

The test of statistical hypotheses

255

So, if T is the appropriate test statistic for the case at hand – and it turns out that, in general, it is – we must ﬁrst eliminate the indeterminacy brought about by q. One possible solution is to estimate the parameter(s) q by some estimating method and use the estimate q˜ in eq. (6.51). At this point, however, two objections come to mind. First, by so doing the probabilities pj are no longer constants but depend on the sample (in fact, no matter which ˜ thus estimation method we choose, the sample must be used to calculate q), implying that eq. (6.48) will probably no longer hold. Second, if eq. (6.48) does not hold but there exists nonetheless a limiting distribution for T, is this limiting distribution independent on the estimation method used to ˜ obtain q? The way out of this rather intricate situation was found in the 1920s by Fisher who showed that for an important class of estimation methods the χ 2 distribution is still the asymptotic distribution of T but eq. (6.48) must be modiﬁed to ˜ → χ 2 (r − k − 1) [D] T(q)

(6.52)

thus determining that the effect of the k unknown parameters is just a decrease – of precisely k units, one unit for each estimated parameter – of the number of degrees of freedom. Such a simple result does not correspond to a simple proof and the interested reader is referred to Chapter 30 of [4] for the details. There, the reader will also ﬁnd the (rather mild) conditions on the continuity and differentiability of the functions pj (q) for eq. (6.52) to hold. On the practical side, once we have obtained the estimate q˜ by means of an appropriate method – we will come to this point shortly – eq. (6.52) implies that the ‘large-sample’ rejection region for the test is 7 8 2 ˜ ≥ χ α;r−k−1 1 = x : t(x; q)

(6.53)

and, as above, the probability of rejecting a true null hypothesis (type I error) is approximately equal to α. In regard to the estimation method, we can argue that a ‘good’ estimate of q can be obtained by making T(q) as small as possible (note that the smaller is T, the better it agrees with the null hypothesis) so that our estimate q˜ can be obtained by solving for the unknowns θ1 , . . . , θk the system of k equations r nj − npj (nj − npj )2 ∂pj − =0 pj ∂θi 2np2j j=1

(6.54a)

where it is understood that pj = pj (q) and i = 1, 2, . . . , k. This is called the χ 2 minimum method of estimation and its advantage is that, under sufﬁciently general conditions, it leads to estimates that are consistent, asymptotically

256 Mathematical statistics normal and asymptotically efﬁcient. Its main drawback, however, is that ﬁnding the solution of (6.54a) is generally a difﬁcult task. For large values of n, fortunately, it can be shown that the second term within parenthesis becomes negligible and therefore the estimate q˜ can be obtained by solving the modiﬁed, and simpler, equations

r nj − npj ∂pj j=1

pj

∂θi

=

r nj ∂pj j=1

pj

∂θi

=0

(6.54b)

where the ﬁrst equality is due to the condition j pj (q) = 1 for all q ∈ . Equation (6.54b) express the so-called ‘modiﬁed χ 2 minimum method’ which, for large samples, gives estimates with the same asymptotic properties as the χ 2 method of eqs (6.54a) and (6.54b). This asymptotic behaviour of the ‘modiﬁed χ 2 estimates’ is not surprising if one notes that, for the observations grouped by means of the partition D1 , D2 , . . . , Dr , q˜ coincides with the ML estimate qˆ g (the subscript g is for ‘grouped’). In fact, once the partition {Dj }rj=1 has been chosen, the r.v.s Nj are distributed according to the multinomial probability law (eq. (3.46a)) because each observation xi can fall in the interval Dj with probability pj . It follows that the likelihood function of the grouped observations Lg (n1 , . . . , nr ; q) is given by

nj n pj (q) n1 ! · · · nr ! r

Lg (N1 = n1 , . . . , Nr = nr ; q) =

(6.55)

j=1

and the ML estimate qˆ g of q is obtained by solving the system of equations ∂ log Lg /∂θi = 0 (i = 1, 2, . . . , k) which, when written explicitly, coincides with (6.54b). At this point an interesting remark can be made. If we calculate the ML estimate before grouping by maximizing the ‘ungrouped’ likelihood function L(x; q) = f (x1 ; q) · · · f (xn ; q) – which, we note, is different from (6.55) and therefore leads to an estimate qˆ = qˆ g – then qˆ is probably a better estimate than qˆ g because it uses the entire information from the sample (grouping, in fact, leads to a partial loss of information). Furthermore, maximizing L(x; q) is often computationally easier than ﬁnding the solution of eq. (6.54b). Unfortunately, while it is true that (eq. (6.52)) T(qˆ g ) → χ 2 (r − k − 1) [D], it ˆ → χ 2 (r − k − 1) [D] does not hold in general and has been shown that T(q) ˆ is more complicated than the χ 2 distrithe asymptotic distribution of T(q) bution with r−k−1 degrees of freedom. In conclusion, we need to group the data ﬁrst and then estimate the unknown parameter(s); by so doing, Cramer [4] shows that the above results are valid for any set of asymptotically normal and asymptotically efﬁcient estimates of the parameters (however obtained, that is, not necessarily by means of the modiﬁed χ 2 method).

The test of statistical hypotheses

257

Example 6.8 Suppose we want to test the null hypothesis that some observed data come from a normal distribution with unknown mean µ = θ1 and variance σ 2 = θ22 so that q = (θ1 , θ2 ) is the set of k = 2 unknown parameters. For j = 1, . . . , r let the grouping intervals be deﬁned as Dj = (xj−1 , xj ] where, in particular, x0 = −∞, xr = ∞ and xj = x1 + (j − 1) d for j = 1, . . . , r−1, x1 and d being appropriate constants (in other words, the Dj s are a partition of the real line in non-overlapping intervals which, except for the ﬁrst and the last, all have a constant width d). Also, in order to simplify the notation, let 1 g(x; q) = √ exp 2π

−

(x − θ1 )2

2θ22

With these deﬁnitions we have the ‘theoretical’ probabilities pj (q) =

1 θ2

g(x; q) dx

(6.56)

Dj

so that, calculating the appropriate derivatives, the modiﬁed minimal χ 2 estimate’ of q is obtained by solving the system of two equations r nj (x − θ1 )g(x; q) dx = 0 pj (q) j=1

Dj

⎛ ⎞ r nj ⎜ ⎟ 2 2 ⎝ (x − θ1 ) g(x; q) dx − θ2 g(x; q) dx⎠ = 0 pj (q) j=1

Dj

Dj

which, after rearranging terms and taking θ1 =

nj j

θ22

n

(6.57a)

nj = n into account, become

xg dx g dx

nj (x − θ1 )2 g dx = n g dx

(6.57b)

j

where all integrals are on Dj and, for brevity, we have omitted the functional dependence of g(x; q). For small values of the interval width d and assuming n1 = nr = 0 (i.e. the extreme intervals contain no data) we can ﬁnd an approximate solution of eq. (6.57b) by replacing each function under integral by its corresponding value at the midpoint ξj of the interval Dj . This leads

258 Mathematical statistics to the ‘grouped’ estimate qˆ g = (θˆ1 , θˆ22 ), where 1 θˆ1 ∼ nj ξj = n r−1 j=2

1 θˆ22 ∼ nj (ξj − θˆ1 )2 = n r−1

(6.58)

j=2

In general, the approximate estimates (6.58) are sufﬁciently good for practical purposes even if the extreme intervals are not empty but contain a small part of the data. So, if y = (y1 , . . . , yn ) is the set of data obtained by the experiment (we call the data yi , and not xi as usual, to avoid confusion with the intervals extreme points deﬁned above) we reject the null hypothesis of normality at the level α if t(qˆ g ) =

[nj − npj (qˆ g )] 2 ≥ χα; r−3 npj (qˆ g )

(6.59)

j

2 2 where χ α; r−3 is the α-upper quantile of the distribution χ (r − 3).

As a numerical example, suppose we have n = 1000 observations of a r.v. which we suspect to be normal; also, let the minimum and maximum observed values be ymin = 36.4 and ymax = 98.3, respectively. Let us choose a partition of the real line in r = 9 intervals with d = 10 and D1 = (−∞, 35], D2 = (35, 45] etc. up to D8 = (95, 105] and D9 = (105, ∞). With this partition, suppose further that our data give the absolute frequencies n = (n1 , . . . , n9 ) = (0, 5, 60, 233, 393, 254, 49, 6, 0)

(6.60)

from which, since the intervals midpoints are ξ2 = 40, ξ3 = 50, . . . , ξ8 = 100, we calculate (eq. (6.58)) the grouped estimates m ˆ = 70.02 and sˆ2 = 102.20 for the mean and the variance, respectively. A normal distribution with this mean and variance leads to the (approximate) set of theoretical frequencies np = n(p2 , . . . , p8 ) ∼ = (4.8, 55.5, 241.5, 394.6, 242.4, 56.0, 4.9)

(6.61)

where the approximation lies in the fact that we calculated the integrals (6.56) by using, once again, the value of the function at the midpoints ξj . Finally, using the experimental and theoretical values of eqs. (6.60) 2 and (6.61) we get t(qˆ g ) = 2.36 which is less than χ 0.05; 6 = 12.59 therefore implying that we accept the null hypothesis of normality.

The test of statistical hypotheses

259

This example is just to illustrate the method and in a real case we should use a ﬁner partition, say r ∼ = 15. In this case we would probably have to pool the extreme intervals to comply with the suggestion npj ≥ 5. So, for instance if we choose d = 5 and the ﬁrst two non-empty intervals have the observed frequencies 1 and 4, we can pool these two intervals to obtain an interval of width d = 10 with a frequency count of 5. In this case Cramer [4] suggests to calculate the estimates with the original grouping (i.e. before pooling), and then perform the test by using the quantiles of the distribution χ 2 (r − 3), where r is the number of intervals after pooling. 6.4.2

Kolmogorov–Smirnov test and some remarks on the empirical distribution function

Similarly to the χ 2 goodness-of-ﬁt test, the Kolmogorov–Smirnov test (KS test) concerns the agreement between an empirical distribution and an assumed theoretical one when this latter is continuous. The test is performed by using the PDFs rather than – as the χ 2 does – the pdfs. The relevant statistic is Dn =

sup

−∞<x

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