Probability Inequalities in Multivariate Distributions

1st Edition - May 28, 1980
Author: Y. L. Tong
Editors: Z. W. Birnbaum, E. Lukas
Language: English
eBook ISBN:
9 7 8 - 1 - 4 8 3 2 - 6 9 2 1 - 4

Probability Inequalities in Multivariate Distributions is a comprehensive treatment of probability inequalities in multivariate distributions, balancing the treatment between… Read more

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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.

PrefaceAcknowledgmentsNotation1 Introduction 1.1. Classification of Probability Inequalities 1.2. Scope and Organization2 Inequalities for Multlvariate Normal Distribution 2.1. Slepian's Inequality 2.2. Multivanate Normal Probabilities of Rectangles 2.3. Other Inequalities for Multivanate Normal Distribution Problems References3 Inequalities for Other Well-Known Distributions 3.1. Multivariate t Distribution 3.2. Multivariate Chi-Square and F Distributions 3.3. An Inequality via Exchangeability Problems References4 Integral Inequalities Over a Symmetric Convex Set 4.1 Anderson's Theorem and Related Results 4.2 Generalizations of Anderson's Theorem 4.3 Inequalities for Elliptically Contoured Distributions Problems References5 Inequalities via Dependence, Association, and Mixture 5.1 Bivariate Dependence 5.2 Association of Random Variables 5.3 Positive Dependence by Mixture of Distributions Problems References6 Inequalities via Majorization and Weak Majorization 6.1 Introduction 6.2 Some Preservation Theorems Under Integral Transforms 6.3 Inequalities via Stochastic Ordering of Random Variables 6.4 Inequalities for Heterogeneous Distributions Problems References7 Distribution-Free Inequalities 7.1 Bonferroni-Type Inequalities 7.2 Chebyshev-Type Inequalities 7.3 Kolmogorov-Type Inequalities Problems References8 Some Applications 8.1 Simultaneous Confidence Regions 8.2 Hypothesis-Testing and Simultaneous Comparisons 8.3 Ranking and Selection Problems 8.4 Reliability and Life Testing Problems ReferencesBibliography A. Books B. Inequalities for Multivariate Normal Distribution C. Inequalities for Multivariate t, Chi-Square, F, and Other Well-Known Distributions D. Integral Inequalities over a Symmetric Convex Set E. Inequalities via Dependence, Association, and Mixture F. Inequalities via Majorization and Weak Majorization G. Distribution-Free Inequalities H. Applications I. Statistical Tables in Multivariate DistributionsAuthor IndexSubject Index