Probability and its Applications
Series Editors Charles Newman Sidney Resnick
For further volumes: www.springer.com/series/4893 B.L.S. Prakasa Rao
Associated Sequences, Demimartingales and Nonparametric Inference B.L.S. Prakasa Rao Department of Mathematics and Statistics University of Hyderabad Hyderabad, India
ISBN 978-3-0348-0239-0 e-ISBN 978-3-0348-0240-6 DOI 10.1007/978-3-0348-0240-6 Springer Basel Dordrecht Heidelberg London New York
Library of Congress Control Number: 2011941605
Mathematics Subject Classification (2010): 60G48
© Springer Basel AG 2012 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained.
Printed on acid-free paper
Springer Basel AG is part of Springer Science+Business Media (www.birkhauser-science.com) In memory of my mother Bhagavatula Saradamba (1926–2002) for her abiding trust in me and for her love and affection Contents
Preface xi
1 Associated Random Variables and Related Concepts 1 1.1 Introduction ...... 1 1.2 Some Probabilistic Properties of Associated Sequences ...... 5 1.3 Related Concepts of Association ...... 23
2 Demimartingales 31 2.1 Introduction ...... 31 2.2 Characteristic Function Inequalities ...... 37 2.3 Doob Type Maximal Inequality ...... 39 2.4 An Upcrossing Inequality ...... 42 2.5 Chow Type Maximal Inequality ...... 44 2.6 Whittle Type Maximal Inequality ...... 50 2.7 More on Maximal Inequalities ...... 54 2.8 Maximal φ-Inequalities for Nonnegative Demisubmartingales . . . 58 2.9 Maximal Inequalities for Functions of Demisubmartingales ..... 65 2.10 Central Limit Theorems ...... 69 2.11 Dominated Demisubmartingales ...... 70
3 N-Demimartingales 73 3.1 Introduction ...... 73 3.2 Maximal Inequalities...... 79 3.3 More on Maximal Inequalities ...... 81 3.4 Downcrossing Inequality ...... 86 3.5 Chow Type Maximal Inequality ...... 89 3.6 Functions of N-Demimartingales ...... 92 3.7 Strong Law of Large Numbers ...... 94 3.8 Azuma Type Inequality ...... 95 3.9 Marcinkiewicz-Zygmund Type inequality ...... 98 3.10 Comparison Theorem on Moment Inequalities ...... 102
vii viii Contents
4 Conditional Demimartingales 105 4.1 Introduction ...... 105 4.2 Conditional Independence ...... 105 4.3 Conditional Association ...... 108 4.4 Conditional Demimartingales ...... 109
5 Multidimensionally Indexed Demimartingales and Continuous Parameter Demimartingales 113 5.1 Introduction ...... 113 5.2 Multidimensionally Indexed Demimartingales ...... 113 5.3 Chow Type Maximal Inequality for Two-Parameter Demimartingales ...... 114 5.4 Continuous Parameter Demisubmartingales ...... 118 5.5 Maximal Inequality for Continuous Parameter Demisubmartingales 119 5.6 Upcrossing Inequality ...... 122
6 Limit Theorems for Associated Random Variables 127 6.1 Introduction ...... 127 6.2 Covariance Inequalities ...... 128 6.3 Hajek-Renyi Type Inequalities ...... 137 6.4 Exponential Inequality ...... 143 6.5 Non-uniform and Uniform Berry-Esseen Type Bounds ...... 145 6.6 Limit Theorems for U-Statistics ...... 150 6.7 More Limit Theorems for U-Statistics ...... 157 6.8 Application to a Two-Sample Problem ...... 172 6.9 Limit Theorems for V -Statistics ...... 180 6.10 Limit Theorems for Associated Random Fields ...... 182 6.11Remarks...... 183
7 Nonparametric Estimation for Associated Sequences 187 7.1 Introduction ...... 187 7.2 Nonparametric Estimation for Survival Function ...... 188 7.3 Nonparametric Density Estimation ...... 194 7.4 Nonparametric Failure Rate Estimation ...... 213 7.5 Nonparametric Mean Residual Life Function Estimation ...... 214 7.6Remarks...... 215
8 Nonparametric Tests for Associated Sequences 217 8.1 Introduction ...... 217 8.2 More on Covariance Inequalities ...... 217 8.3 Tests for Location ...... 220 8.4 Mann-Whitney Test ...... 227 Contents ix
9 Nonparametric Tests for Change in Marginal Density Function for Associated Sequences 235 9.1 Introduction ...... 235 9.2 Tests for Change Point ...... 236 9.3 Test for Change in Marginal Density Function ...... 242
Bibliography 253
Index 271
Preface
One of the basic aims of theory of probability and statistics is to build stochastic models which explain the phenomenon under investigation and explore the depen- dence among various covariates which influence this phenomenon. Classic examples are the concepts of Markov dependence or of mixing for random processes. Esary, Proschan and Walkup introduced the concept of association for random variables. In several situations, for example, in reliability and survival analysis, the random variables of lifetimes for components are not independent but associated. Newman and Wright introduced the notion of a demimartingale whose properties extend those of a martingale. It can be shown that the partial sums of mean zero associ- ated random variables form a demimartingale. The study of demimartingales and related concepts is the subject matter of this book. Along with a discussion on demimartingales and related concepts, we review some recent results on proba- bilistic inequalities for sequences of associated random variables and methods of nonparametric inference for such sequences. The idea to write this book occurred following the invitation of Professor Tasos Christofides to me to visit the University of Cyprus, Nicosia in October 2010. Professor Tasos Christofides has contributed extensively to the subject mat- ter of this book. It is a pleasure to thank him for his invitation. Professor Isha Dewan of the Indian Statistical Institute and I have been involved in the study of probabilistic properties of associated sequences and in developing nonparametric inference for such processes during the years 1990-2004 while I was at the Indian Statistical Institute. This book is a culmination of those efforts and I would like to thank Professor Isha Dewan for her collaboration over the years. Thanks are due to my wife Vasanta Bhagavatula for her continued support to pursue my academic interests even after my retirement.
B.L.S. Prakasa Rao Hyderabad, July 1, 2011
xi
Chapter 1
Associated Random Variables and Related Concepts
1.1 Introduction
In classical statistical inference, the observed random variables of interest are gen- erally assumed to be independent and identically distributed. However in some real life situations, the random variables need not be independent. In reliability studies, there are structures in which the components share the load, so that failure of one component results in increased load on each of the remaining components. Mini- mal path structures of a coherent system having components in common behave in a ‘similar’ manner. Failure of a component will adversely effect the performance of all the minimal path structures containing it. In both the examples given above, the random variables of interest are not independent but are “associated” , a concept we will define soon. This book is concerned with the study of properties of stochastic processes termed as demimartingales and N-demimartingales and related concepts. As we will see in the next chapter, an important example of a demimartingale is the sequence of partial sums of mean zero associated ran- dom variables. We will now briefly review some properties of associated random variables. We will come back to the study of these sequences again in Chapter 6. We assume that all the expectations involved in the following discussions exist. Hoeffding (1940) (cf. Lehmann (1966)) proved the following result. Theorem 1.1.1. Let (X, Y ) be a bivariate random vector such that E(X2) < ∞ and E(Y 2) < ∞.Then ∞ ∞ Cov(X, Y )= H(x, y)dxdy (1.1.1) −∞ −∞
B.L.S. Prakasa Rao, Associated Sequences, Demimartingales 1 and Nonparametric Inference, Probability and its Applications, DOI 10.1007/978-3-0348-0240-6_1, © Springer Basel AG 2012 2 Chapter 1. Associated Random Variables and Related Concepts where,
H(x, y)=P [X>x,Y>y] − P [X>x]P [Y>y] = P [X ≤ x, Y ≤ y] − P [X ≤ x]P [Y ≤ y]. (1.1.2)
Proof. Let (X1,Y1)and(X2,Y2) be independent and identically distributed ran- dom vectors. Then,
2[E(X1Y1) − E(X1)E(Y1)]
= E[(X1 − X2)(Y1 − Y2)] ∞ ∞ = E[ [I(u, X1) − I(u, X2)][I(v, Y1) − I(v, Y2)]dudv] −∞ −∞ ∞ ∞ = E([I(u, X1) − I(u, X2)][I(v, Y1) − I(v, Y2)])dudv −∞ −∞ where I(u, a)=1ifu ≤ a and 0 otherwise. The last equality follows as an application of Fubini’s theorem. Relation (1.1.1) is known as the Hoeffding identity . A generalized Hoeffding identity has been proved by Block and Fang (1983) for multidimensional random vectors. Newman (1980) showed that for any two functions h(·)andg(·)with E[h(X)]2 <∞ and E[g(Y )]2 < ∞ and finite derivatives h(·)andg(·), ∞ ∞ Cov(h(X),g(Y )) = h(x)g(y)H(x, y)dx dy (1.1.3) −∞ −∞ hereinafter called Newman’s identity. Yu (1993) extended the relation (1.1.3) to even-dimensional random vectors. Prakasa Rao (1998) further extended this iden- tity following Queseda-Molina (1992). As a departure from independence, a bivariate notion of positive quadrant dependence was introduced by Lehmann (1966). Definition. A pair of random variables (X, Y ) is said to be positively quadrant dependent (PQD) if
P [X ≤ x, Y ≤ y] ≥ P [X ≤ x]P [Y ≤ y] ∀ x, y (1.1.4) or equivalently H(x, y) ≥ 0,x,y∈ R. (1.1.5) It can be shown that the condition (1.1.5) is equivalent to the following: for any pair of nondecreasing functions h and g on R,
Cov(h(X),g(Y )) ≥ 0. (1.1.6) 1.1. Introduction 3
A stronger condition is that, for a pair of random variables (X, Y ) and for any two real componentwise nondecreasing functions h and g on R2,
Cov(h(X, Y ),g(X, Y )) ≥ 0. (1.1.7)
As a natural multivariate extension of (1.1.7), the following concept of asso- ciation was introduced by Esary, Proschan and Walkup (1967).
Definition. A finite collection of random variables {Xj,1≤ j ≤ n} is said to be associated if for every every choice of componentwise nondecreasing functions h and g from Rn to R,
Cov(h(X1,...,Xn),g(X1,...,Xn)) ≥ 0 (1.1.8) whenever it exists; an infinite collection of random variables {Xn, n ≥ 1} is said to be associated if every finite sub-collection is associated. It is easy to see that any set of independent random variables is associated (cf. Esary, Proschan and Walkup (1967)). Associated random variables arise in widely different areas such as reliability, statistical mechanics, percolation theory etc. The basic concept of association has appeared in the context of percolation models (cf. Harris (1960)) and later applied to the Ising models in statistical mechanics in Fortuin et al. (1971) and Lebowitz (1972). Examples. (i) Let {Xi, i ≥ 1} be independent random variables and Y be an- other random variable independent of {Xi, i ≥ 1}.Then the random variables {Xi = Xi + Y , i ≥ 1} are associated. Thus, independent random variables subject to the same stress are associated (cf. Barlow and Proschan (1975)). For an application of this observation to modelling dependent competing risks, see Bagai and Prakasa Rao (1992). (ii) Order statistics corresponding to a finite set of independent random variables are associated.
(iii) Let {Xi,1≤ i ≤ n} be associated random variables. Let Si = X1 + ···+ Xi, 1 ≤ i ≤ n. Then the set {Si,1≤ i ≤ n} is associated. (iv) Positively correlated normal random variables are associated (cf. Pitt (1982)).
(v) Suppose a random vector (X1,...,Xm) has a multivariate exponential dis- tribution F (x1,...,xm)( cf. Marshall and Olkin (1967)) with
1 − F (x1,...,xm) m =exp[− λixi − λij max(xi,xj) i=1 i