Approximate Distributions of Order Statistics: With Applications to Nonparametric Statistics

362 Pages·1989·9.66 MB·English

by  R.-D. Reiss (auth.)

Springer Series in Statistics
Advisors:
J. Berger, S. Fienberg, J. Gani,
K. Krickeberg, B. Singer
Springer Series in Statistics
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Farrell: Multivariate Calculation.
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Hartigan: Bayes Theory.
Heyer: Theory of Statistical Experiments.
Jolliffe: Principal Component Analysis.
Kres: Statistical Tables for Multivariate Analysis.
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Le Cam: Asymptotic Methods in Statistical Decision Theory.
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Reiss: Approximate Distributions of Order Statistics: With Applications to
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R.-D. Reiss
Approximate Distributions
of Order Statistics
With Applications to N onparametric
Statistics
With 30 Illustrations
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo
R.-D. Reiss
Universitat Gesamthochschule Siegen
Fachbereich 6, Mathematik
D-5900 Siegen
Federal Republic of Germany
Mathematics Subject Classification (1980): 62-07, 62B15, 62E20, 62G05, 62G10, 62G30
Library of Congress Cataloging-in-Publication Data
Reiss, Rolf-Dieter.
Approximate distributions of order statistics.
(Springer series in statistics)
Bibliography: p.
Includes indexes.
1. Order statistics. 2. Asymptotic distribution
(Probability theory) 3. Nonparametric statistics.
I. Title. II. Series.
QA278.7.R45 1989 519.5 88-24844
Printed on acid-free paper.
© 1989 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1s t edition 1989
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9 8 7 6 5 4 3 2 1
ISBN-13:978-1-4613-9622-2 e-ISBN-13:978-1-4613-9620-8
DOl: 10.1007/978-1-4613-9620-8
To Margit, Maximilian, Cornelia, and Thomas
Preface
This book is designed as a unified and mathematically rigorous treatment of
some recent developments of the asymptotic distribution theory of order
statistics (including the extreme order statistics) that are relevant for statistical
theory and its applications. Particular emphasis is placed on results concern
ing the accuracy oflimit theorems, on higher order approximations, and other
approximations in quite a general sense.
Contrary to the classical limit theorems that primarily concern the weak
convergence of distribution functions, our main results will be formulated in
terms of the variational and the Hellinger distance. These results will form the
proper springboard for the investigation of parametric approximations of
nonparametric models of joint distributions of order statistics. The approxi
mating models include normal as well as extreme value models. Several
applications will show the usefulness of this approach.
Other recent developments in statistics like nonparametric curve estima
tion and the bootstrap method will be studied as far as order statistics are
concerned. 1n connection with this, graphical methods will, to some extent,
be explored.
The prerequisite for handling the indicated problems is a profound knowl
edge of distributional properties of order statistics. Thus, we collect several
basic tools (of finite and asymptotic nature) that are either scattered in litera
ture or are not elaborated to such an extent that would satisfy our present
requirements. For example, the Markov property of order statistics is studied
in detail. This part of the book that has the characteristics of a textbook is
supplemented by several well-known results.
The book is intended for students and research workers in probability and
statistics, and practitioners involved in applications of mathematical results
concerning order statistics and extremes. The knowledge of standard calculus
viii Preface
and topics that are taught in introductory probability and statistics courses
are necessary for the understanding of this book. To reinforce previous knowl
edge as well as to fill gaps, we shall frequently give a short exposition of
probabilistic and statistical concepts (e.g., that of conditional distribution and
approximate sufficiency).
The results are often formulated for distributions themselves (and not only
for distribution functions) and so we need, as far as order statistics are
concerned, the notion of Borel sets in a Euclidean space. Intervals, open sets,
and closed sets are special Borel sets. Large parts of this book can be under
stood without prior knowledge of technical details of measure-theoretic
nature.
My research work on order statistics started at the University of Cologne,
where influenced by J. Pfanzagl, I became familiar with expansions and
statistical problems. Lecture notes of a course on order statistics held at the
University of Freiburg during the academic year 1976/77 can be regarded as
an early forerunner of the book.
I would like to thank my students B. Dohmann, G. Heer, and E. Kaufmann
for their programming assistance. G. Heer also skillfully read through larger
parts of the manuscript. It gives me great pleasure to acknowledge the co
operation, documented by several articles, with my colleague M. Falk. The
excellent atmosphere within the small statistical research group at the Univer
sity of Siegen, and including A. Janssen and F. Marohn, facilitated the writing
of this book. Finally, I would like to thank W. Stute, and those not mentioned
individually, for their comments.
Siegen, FR Germany Rolf-Dieter Reiss
Contents
Preface vii
CHAPTER 0
Introduction
0.1. Weak and Strong Convergence 1
0.2. Approximations 3
0.3. The Role of Order Statistics in Nonparametric Statistics 4
0.4. Central and Extreme Order Statistics 5
0.5. The Restriction to Independent and Identically Distributed
Random Variables 6
0.6. Graphical Methods 6
0.7. A Guide to the Contents 7
0.8. Notation and Conventions 8
PART I
Exact Distributions and Basic Tools
CHAPTER 1
Distribution Functions, Densities, and Representations 11
1.1. Introduction to Basic Concepts 11
1.2. The Quantile Transformation 14
1.3. Single Order Statistics, Extremes 20
1.4. Joint Distribution of Several Order Statistics 27
1.5. Extensions to Continuous and Discontinuous Distribution Functions 32
1.6. Spacings, Representations, Generalized Pareto Distribution Functions 36
1. 7. Moments, Modes, and Medians 44
1.8. Conditional Distributions of Order Statistics 51
P.1. Problems and Supplements 56
Bibliographical Notes 61
x Contents
CHAPTER 2
Multivariate Order Statistics 64
2.1. Introduction 64
2.2. Distribution Functions and Densities 68
P.2. Problems and Supplements 78
Bibliographical Notes 81
CHAPTER 3
Inequalities and the Concept of Expansions 83
3.1. Inequalities for Distributions of Order Statistics 83
3.2. Expansions of Finite Length 89
3.3. Distances of Measures: Convergence and Inequalities 94
P.3. Problems and Supplements 102
Bibliographical Notes 104
PART II
Asymptotic Theory
CHAPTER 4
Approximations to Distributions of Central Order Statistics 107
4.1. Asymptotic Normality of Central Sequences 108
4.2. Expansions: A Single Central Order Statistic 114
4.3. Asymptotic Independence from the Underlying Distribution
Function 123
4.4. The Approximate Multivariate Normal Distribution 129
4.5. Asymptotic Normality and Expansions of Joint Distributions 131
4.6. Expansions of Distribution Functions of Order Statistics 138
4.7. Local Limit Theorems and Moderate Deviations 142
P.4. Problems and Supplements 145
Bibliographical Notes 148
CHAPTER 5
Approximations to Distributions of Extremes 151
5.1. Asymptotic Distributions of Extreme Sequences 152
5.2. Hellinger Distance between Exact and Approximate Distributions
of Sample Maxima 164
5.3. The Structure of Asymptotic Joint Distributions of Extremes 176
5.4. Expansions of Distributions of Extremes of Generalized Pareto
Random Variables 181
5.5. Variational Distance between Exact and Approximate
Joint Distributions of Extremes 186
5.6. Variational Distance between Empirical and Poisson Processes 190
P.5. Problems and Supplements 194
Bibliographical Notes 201
Contents xi
CHAPTER 6
Other Important Approximations 206
6.1. Approximations of Moments and Quantiles 206
6.2. Functions of Order Statistics 209
6.3. Bahadur Approximation 216
6.4. Bootstrap Distribution Function of a Quantile 220
P.6. Problems and Supplements 226
Bibliographical Notes 227
CHAPTER 7
Approximations in the Multivariate Case 229
7.1. Asymptotic Normality of Central Order Statistics 229
7.2. Multivariate Extremes 232
P.7. Problems and Supplements 237
Bibliographical Notes 238
PART III
Statistical Models and Procedures
CHAPTER 8
Evaluating the Quantile and Density Quantile Function 243
8.1. Sample Quantiles 243
8.2. Kernel Type Estimators of Quantiles 248
8.3. Asymptotic Performance of Quantile Estimators 260
8.4. Bootstrap via Smooth Sample Quantile Function 265
P.8. Problems and Supplements 268
Bibliographical Notes 270
CHAPTER 9
Extreme Value Models 272
9.1. Some Basic Concepts of Statistical Theory 273
9.2. Efficient Estimation in Extreme Value Models 276
9.3. Semiparametric Models for Sample Maxima 279
9.4. Parametric Models Belonging to Upper Extremes 281
9.5. Inference Based on Upper Extremes 283
9.6. Comparison of Different Approaches 284
9.7. Estimating the Quantile Function Near the Endpoints 286
P.9. Problems and Supplements 289
Bibliographical Notes 290
CHAPTER 10
Approximate Sufficiency of Sparse Order Statistics 292
10.1. Comparison of Statistical Models via Markov Kernels 292
10.2. Approximate Sufficiency over a Neighborhood of a Fixed
Distribution 299