Yung-Liang T. Probability Inequalities in Multivariate Distributions

New York: Academic Press, 1980. — 255 p.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.Notation
Classification of Probability Inequalities
Scope and Organization
Inequalities for Multivariate Normal Distribution
Slepian’s Inequality
Multivariate Normal Probabilities of Rectangles
Other Inequalities for Multivariate Normal Distribution
ProblemsMultivariate t Distribution
Multivariate Chi-Square and F Distributions
An Inequality via Exchangeability
ProblemsAnderson’s Theorem and Related Results
Generalizations of Anderson’s Theorem
Inequalities for Elliptically Contoured Distributions
ProblemsInequalities via Dependence, Association, and Mixture
Bivariate Dependence
Association of Random Variables
Positive Dependence by Mixture of Distributions
ProblemsSome Preservation Theorems under Integral Transforms
Inequalities via Stochastic Ordering of Random Variables
Inequalities for Heterogeneous Distributions
ProblemsDIstrlbutlon-Free Inequalities
Bonferroni-Type Inequalities
Chebyshev-Type Inequalities
Kolmogorov-Type Inequalities
ProblemsSimultaneous Confidence Regions
Hypothesis-Testing and Simultaneous Comparisons
Ranking and Selection Problems
Reliability and Life Testing
ProblemsBooks
Inequalities for Multivariate Normal Distribution
Inequalities for Multivariate t, Chi-Square, F,and Other Well-Known Distributions
Integral Inequalities over a Symmetric Convex Set
Inequalities via Dependence, Association, and Mixture
Inequalities via Majorization and Weak Majorization
Distribution-Free Inequalities
Applications
Statistical Tables in Multivariate Distributions
Author Index
Subject Index