The Multivariate Normal Distribution

280 Pages·1990·20.452 MB·English

by  Y. L. Tong (auth.)

Springer Series in Statistics
Advisors:
D. Brillinger, S. Fienberg, J. Gani,
J. Hartigan, K. Krickeberg
Springer Series in Statistics
Andrews/Henberg: Data: A Collection of Problems from Many Fields for the
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Anscombe: Computing in Statistical Science through APL.
Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition.
Bremaud: Point Processes and Queues: Martingale Dynamics.
BrockwelljDavis: Time Series: Theory and Methods.
DaleyjVere-lones: An Introduction to the Theory of Point Processes.
Dzhaparidze: Parameter Estimation and Hypothesis Testing in Spectral Analysis
of Stationary Time Series.
Farrell: Multivariate Calculation.
GoodmanjKJUskal: Measures of Association for Cross Classifications.
Hartigan: Bayes Theory.
Heyer: Theory of Statistical Experiments.
lolliffe: Principal Component Analysis.
Kres: Statistical Tables for Multivariate Analysis.
Leadbetter/LindgrenjRootzen: Extremes and Related Properties of Random
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Le Cam: Asymptotic Methods in Statistical Decision Theory.
Manoukian: Modern Concepts and Theorems of Mathematical Statistics.
Miller, II'.: Simulaneous Statistical Inference, 2nd edition.
Moste/lerjWa/lace: Applied Bayesian and Classical Inference: The Case of The
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Pollard: Convergence of Stochastic Processes.
Pmu/Gibbolls: Concepts of Nonparametric Theory.
Read/Cressie: Goodness-of-Fit Statistics for Discrete Multivariate Data.
Reiss: Approximate Distributions of Order Statistics: With Applications to
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Sachs: Applied Statistics: A Handbook of Techniques, 2nd edition.
Seneta: Non-Negative Matrices and Markov Chains.
Siegmund: Sequential Analysis: Tests and Confidence Intervals.
Tong: The Multivariate Normal Distribution
Vapnik: Estimation of Dependences Based on Empirical Data.
West/Hamson: Bayesian Forecasting and Dynamic Models
Wolter: Introduction to Variance Estimation.
Yag/om: Correlation Theory of Stationary and Related Random Functions I:
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Supplementary Notes and References.
Y.L. Tong
The Multivariate Normal
Distribution
Springer-Verlag
New York Berlin Heidelberg
London Paris Tokyo Hong Kong
Y.L. Tong
School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160
U.S.A.
AMS Mathematics Subject Classifications (1980): 60E05, 62H99
Library of Congress Cataloging in Publication Data
Tong, Y.L. (Yung Liang), 1935-
The multivariate normal distribution I Y.L. Tong.
P. cm. - (Springer series in statistics)
Includes bibliographical references.
ISBN-13:978-1-4613-9657-4 (alk. paper)
1. Distribution (Probability theory) 2. Multivariate analysis.
1. Title. II. Series.
QA273.6.T67 1990
519.2'4-dc20 89-21929
CIP
Printed on acid-free paper
© 1990 by Springer-Verlag New York Inc.
Softcover reprint of the hardcover 1st edition 1990
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Typeset by Asco Trade Typesetting Ltd., Hong Kong.
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ISBN-13:978-1-4613-9657-4 e-ISBN-13:978-1-4613-9655-0
DOl: to.1 007/978-1-4613-9655-0
To My Family
Contents
Preface Xl
Basic Notation and Numbering System Xlll
CHAPTER 1
Introduction 1
1.1. Some Fundamental Properties 1
1.2. Historical Remarks 2
1.3. Characterization 3
1.4. Scope and Organization 3
CHAPTER 2
The Bivariate Normal Distribution 6
2.1. Some Distribution Properties 7
2.2. The Distribution Function and Sampling Distributions 14
2.3. Dependence and the Correlation Coefficient 19
Problems 21
CHAPTER 3
Fundamental Properties and Sampling Distributions of
the Multivariate Normal Distribution 23
3.1. Preliminaries 23
3.2. Definitions of the Multivariate Normal Distribution 26
3.3. Basic Distribution Properties 30
3.4. Regression and Correlation 35
3.5. Sampling Distributions 47
Problems 59
viii Conterits
CHAPTER 4
Other Related Properties 62
4.1. The Elliptically Contoured Family of Distributions and
the Multivariate Normal 62
4.2. Log-Concavity and Unimodality Properties 68
4.3. MTP2 and MRR2 Properties 73
4.4. Schur-Concavity Property 79
4.5. Arrangement-Increasing Property 84
Problems 89
CHAPTER 5
Positively Dependent and Exchangeable Normal Variables 91
5.1. Positively Dependent Normal Variables 92
5.2. Permutation-Symmetric Normal Variables 104
5.3. Exchangeable Normal Variables 108
Problems 120
CHAPTER 6
Order Statistics of Normal Variables 123
6~1. Order Statistics of Exchangeable Normal Variables 123
6.2. Positive Dependence of Order Statistics of Normal Variables 130
6.3. Distributions of Certain Partial Sums and Linear
Combinations of Order Statistics 136
6.4. Miscellaneous Results 140
Problems 147
CHAPTER 7
Related Inequalities 150
7.1. Introduction 150
7.2. Dependence-Related Inequalities 152
7.3. Dimension-Related Inequalities 154
7.4. Probability Inequalities for Asymmetric Geometric Regions 161
7.5. Other Related Inequalities 169
Problems 177
CHAPTER 8
Statistical Computing Related to the Multivariate
Normal Distribution 181
8.1. Generation of Multivariate Normal Variates 181
8.2. Evaluation and Approximations of Multivariate Normal
Probability Integrals 186
8.3. Computation of One-Sided and Two-Sided Multivariate
Normal Probability Integrals 193
8.4. The Tables 194
Problems 199
Contents ix
CHAPTER 9
The Multivariate t Distribution 202
9.1. Distribution Properties 204
9.2. Probability Inequalities 207
9.3. Convergence to the Multivariate Normal Distribution 211
9.4. Tables for Exchangeable t Variables 213
Problems 216
References 219
Appendix-Tables 229
Author Index 261
Subject Index 265
Preface
The multivariate normal distribution has played a predominant role in the
historical development of statistical theory, and has made its appearance in
various areas of applications. Although many of the results concerning the
multivariate normal distribution are classical, there are important new results
which have been reported recently in the literature but cannot be found in
most books on multivariate analysis. These results are often obtained by
showing that the multivariate normal density function belongs to certain large
families of density functions. Thus, useful properties of such families immedi
ately hold for the multivariate normal distribution.
This book attempts to provide a comprehensive and coherent treatment of
the classical and new results related to the multivariate normal distribution.
The material is organized in a unified modern approach, and the main themes
are dependence, probability inequalities, and their roles in theory and applica
tions. Some general properties of a multivariate normal density function are
discussed, and results that follow from these properties are reviewed exten
sively. The coverage is, to some extent, a matter of taste and is not intended
to be exhaustive, thus more attention is focused on a systematic presentation
of results rather than on a complete listing of them.
Most of the classical results on distribution theory, sampling distributions,
and correlation analysis are presented in Chapters 2 and 3. Chapter 4 deals
with the log-concavity, unimodality, total positivity, Schur-concavity, and
arrangement increasing properties of a multivariate normal density function
and related results. Notions of dependence and their application to the multi
variate normal distribution are discussed in Chapter 5; not surprisingly, the
results involve the covariance matrix of the distribution. Chapter 6 includes
distribution theory and dependence results for the order statistics of normal
variables. Chapter 7 contains inequalities and bounds for the multivariate
xii Preface
normal distribution, including dependence-related inequalities, dimension
related inequalities, and inequalities for the probability contents of geometric
regions in a certain class. Problems on statistical computing, mainly the
generation of multivariate normal variates and the evaluation of multivariate
normal probability integrals, are treated in Chapter 8; tables of equicoordinate
one-sided and two-sided percentage points and probability integrals for ex
changeable normal variables are given in the Appendix. A short chapter
(Chapter 9) on the multivariate t distribution presents results concerning
related distribution theory and convergence to the multivariate normal distri
bution. Chapters 2-9 contain sets of complementary problems. Finally, a
combined list of references can be found at the end of the volume.
This book assumes a basic knowledge of matrix algebra and mathematical
statistics at the undergraduate level, and is accessible to graduate students and
advanced uhndergraduate students in statistics, mathematics, and related
applied areas. Although it is not intended as a textbook, it can be used as
a main reference in a course on multivariate analysis. And, of course, it can
be used as a reference book on the multivariate normal distribution by
researchers.
This work was partially supported by National Science Foundation grants
DMS-8502346 and DMS-8801327 at Georgia Institute of Technology. Need
less to say, I am indebted to the extensive literature in related areas. Professors
Theodore W. Anderson, Herbert A. David, Kai-Tai Fang, Kumar Joag-Dev,
Mark E. Johnson, Samuel Kotz, and Moshe Shaked read all or parts of the
manuscript, and their comments and suggestions resulted in numerous signifi
cant improvements. However, I am solely responsible for errors and omissions.
I am grateful to Professors Ingram Olkin and Frank Proschan for their
inspiration, continuing encouragement, and constructively critical comments,
and to Professor Milton Sobel for his strong influence on my work concerning
the multivariate t distribution. I wish to thank Ms. Annette Rohrs for her
skillful typing and wonderful cooperation, and also, the staff at Springer
Verlag for the neat appearance of the volume. Finally, I thank my wife
Ai-Chuan and our children Frank, Betty, and Lily for their understanding
and support. Frank read Chapter 1 and made some helpful comments, and
Betty spent many long hours with me at the office.
Atlanta, Georgia Y.L. TONG
November 1988