Probability Inequalities in Multivariate Distributions

✍ Scribed by Tong Yung-Liang

• Publisher: Academic Press
• Year: 1980
• Tongue: English
• Leaves: 255
• Series: Probability and Mathematical Statistics
• Category: Library

✦ Synopsis

The area of probability inequalities in multivariate distributions is certainly not new. However, it has experienced a remarkable growth and development during the past two decades or so. Today the subject plays an important role in many areas of statistics and probability, and it presents a very challenging and attractive field of research. On the one hand the theory is beautiful and elegant, and on the other hand the applications of the inequalities seem unlimited. I developed my own interest in this area more than ten years ago as a “user” of inequalities. Later I found the theory extremely fascinating in itself. Consequently, I felt the need for a comprehensive treatment of probability inequalities that would make them easily accessible. It is my hope that this book will serve that purpose.

✦ Table of Contents

Tong Yung-Liang.Probability Inequalities in Multivariate Distributions (Probability and mathematical statistics)(AP,1980)(ISBN 0126949506)(600)(255p) ......Page 4
Copyright ......Page 5
Contents v ......Page 6
Preface ix ......Page 9
Acknowledgments xi ......Page 11
Notation xiii ......Page 13
1.1.Classification of Probability Inequalities 1 ......Page 14
1.2.Scope and Organization 5 ......Page 18
2 Inequalities for Multivariate Normal Distribution 7 ......Page 20
2.1.Slepian’s Inequality 8 ......Page 21
2.2.Multivariate Normal Probabilities of Rectangles 12 ......Page 25
2.3.Other Inequalities for Multivariate Normal Distribution 25 ......Page 38
Problems 31 ......Page 44
References 34 ......Page 47
3.1.Multivariate t Distribution 36 ......Page 49
3.2.Multivariate Chi-Square and F Distributions 40 ......Page 53
3.3.An Inequality via Exchangeability 44 ......Page 57
Problems 47 ......Page 60
References 48 ......Page 61
4.1.Anderson’s Theorem and Related Results 50 ......Page 63
4.2.Generalizations of Anderson’s Theorem 60 ......Page 73
4.3.Inequalities for Elliptically Contoured Distributions 64 ......Page 77
Problems 74 ......Page 87
References 75 ......Page 88
5 Inequalities via Dependence, Association, and Mixture 77......Page 90
5.1.Bivariate Dependence 78 ......Page 91
5.2.Association of Random Variables 85 ......Page 98
5.3.Positive Dependence by Mixture of Distributions 94 ......Page 107
Problems 98 ......Page 111
References 100 ......Page 113
6.1.Introduction 102 ......Page 115
6.2.Some Preservation Theorems under Integral Transforms 105 ......Page 118
6.3.Inequalities via Stochastic Ordering of Random Variables 121 ......Page 134
6.4.Inequalities for Heterogeneous Distributions 131 ......Page 144
Problems 138 ......Page 151
References 140 ......Page 153
7 DIstrlbutlon-Free Inequalities 142......Page 155
7.1.Bonferroni-Type Inequalities 143 ......Page 156
7.2.Chebyshev-Type Inequalities 150 ......Page 163
7.3.Kolmogorov-Type Inequalities 158 ......Page 171
Problems 160 ......Page 173
References 162 ......Page 175
8.1.Simultaneous Confidence Regions 164 ......Page 177
8.2.Hypothesis-Testing and Simultaneous Comparisons 177 ......Page 190
8.3.Ranking and Selection Problems 189 ......Page 202
8.4.Reliability and Life Testing 198 ......Page 211
Problems 202 ......Page 215
References 204 ......Page 217
A.Books 208 ......Page 221
B.Inequalities for Multivariate Normal Distribution 209......Page 222
C.Inequalities for Multivariate t, Chi-Square, F,and Other Well-Known Distributions 211 ......Page 224
D.Integral Inequalities over a Symmetric Convex Set 212 ......Page 225
E.Inequalities via Dependence, Association, and Mixture 214 ......Page 227
F.Inequalities via Majorization and Weak Majorization 216 ......Page 229
G.Distribution-Free Inequalities 219 ......Page 232
H.Applications 220 ......Page 233
I.Statistical Tables in Multivariate Distributions 229 ......Page 242
Author Index 233 ......Page 246
Subject Index 236 ......Page 249
cover......Page 1