Tong / Birnbaum / Lukas Probability Inequalities in Multivariate Distributions

1. Auflage 2014
ISBN: 978-1-4832-6921-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 255 Seiten, Web PDF

Probability Inequalities in Multivariate Distributions is a comprehensive treatment of probability inequalities in multivariate distributions, balancing the treatment between theory and applications. The book is concerned only with those inequalities that are of types T1-T5. The conditions for such inequalities range from very specific to very general. Comprised of eight chapters, this volume begins by presenting a classification of probability inequalities, followed by a discussion on inequalities for multivariate normal distribution as well as their dependence on correlation coefficients. The reader is then introduced to inequalities for other well-known distributions, including the multivariate distributions of t, chi-square, and F; inequalities for a class of symmetric unimodal distributions and for a certain class of random variables that are positively dependent by association or by mixture; and inequalities obtainable through the mathematical tool of majorization and weak majorization. The book also describes some distribution-free inequalities before concluding with an overview of their applications in simultaneous confidence regions, hypothesis testing, multiple decision problems, and reliability and life testing. This monograph is intended for mathematicians, statisticians, students, and those who are primarily interested in inequalities.

Autoren/Hrsg.

Tong, Y. L.

Birnbaum, Z. W.

Lukas, E.

Inhaltsverzeichnis

1;Front Cover;1
2;Probability Inequalities in Multivariate Distributions;4
3;Copyright Page;5
4;Table of Contents;6
5;Preface;10
6;Acknowledgments;12
7;Notation;14
8;Chapter 1. Introduction;16
8.1;1.1. Classification of Probability Inequalities;16
8.2;1.2. Scope and Organization;20
9;Chapter 2. Inequalities for Multlvariate Normal Distribution;22
9.1;2.1. Slepian's Inequality;23
9.2;2.2. Multivanate Normal Probabilities of Rectangles;27
9.3;2.3. Other Inequalities for Multivanate Normal Distribution;40
9.4;Problems;46
9.5;References;49
10;Chapter 3. Inequalities for Other Well-Known Distributions;51
10.1;3.1. Multivariate t Distribution;51
10.2;3.2. Multivariate Chi-Square and F Distributions;55
10.3;3.3. An Inequality via Exchangeability;59
10.4;Problems;62
10.5;References;63
11;Chapter 4. Integral Inequalities over a Symmetric Convex Set;65
11.1;4.1.
Anderson's Theorem and Related Results;65
11.2;4.2.
Generalizations of Anderson's Theorem;75
11.3;4.3.
Inequalities for Elliptically Contoured Distributions;79
11.4;Problems;89
11.5;References;90
12;Chapter 5. Inequalities via Dependence, Association, and Mixture;92
12.1;5.1.
Bivariate Dependence;93
12.2;5.2.
Association of Random Variables;100
12.3;5.3.
Positive Dependence by Mixture of Distributions;109
12.4;Problems;113
12.5;References;115
13;Chapter 6. Inequalities via Majorization and Weak Majorization;117
13.1;6.1.
Introduction;117
13.2;6.2.
Some Preservation Theorems under Integral Transforms;120
13.3;6.3. Inequalities via Stochastic Ordering of Random Variables;136
13.4;6.4. Inequalities for Heterogeneous Distributions;146
13.5;Problems;153
13.6;References;155
14;Chapter 7. Distribution-Free Inequalities;157
14.1;7.1.
Bonferroni-Type Inequalities;158
14.2;7.2.
Chebyshev-Type Inequalities;165
14.3;7.3.
Kolmogorov-Type Inequalities;173
14.4;Problems;175
14.5;References;177
15;Chapter 8. Some Applications;179
15.1;8.1.
Simultaneous Confidence Regions;179
15.2;8.2.
Hypothesis-Testing and Simultaneous Comparisons;192
15.3;8.3.
Ranking and Selection Problems;204
15.4;8.4.
Reliability and Life Testing;213
15.5;Problems;217
15.6;References;219
16;Bibliography;223
16.1;A. Books;223
16.2;B. Inequalities for Multivariate Normal Distribution;224
16.3;C. Inequalities for Multivariate t, Chi-Square, F, and Other Well-Known Distributions;226
16.4;D. Integral Inequalities over a Symmetric Convex Set;227
16.5;E. Inequalities via Dependence, Association, and Mixture;229
16.6;F. Inequalities via Majorization and Weak Majorization;231
16.7;G. Distribution-Free Inequalities;234
16.8;H. Applications;235
16.9;I. Statistical Tables in Multivariate Distributions;244
17;Author Index;248
18;Subject Index;251