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 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.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.Multivariate t Distribution.
Multivariate Chi-Square and F Distributions.
An Inequality via Exchangeability.Anderson’s Theorem and Related Results.
Generalizations of Anderson’s Theorem.
Inequalities for Elliptically Contoured Distributions.Inequalities via Dependence, Association, and Mixture.
Bivariate Dependence.
Association of Random Variables.
Positive Dependence by Mixture of Distributions.Some Preservation Theorems under Integral Transforms.
Inequalities via Stochastic Ordering of Random Variables.
Inequalities for Heterogeneous Distributions.DIstrlbutlon-Free Inequalities.
Bonferroni-Type Inequalities.
Chebyshev-Type Inequalities.
Kolmogorov-Type Inequalities.Simultaneous Confidence Regions.
Hypothesis-Testing and Simultaneous Comparisons.
Ranking and Selection Problems.
Reliability and Life Testing.BooksInequalities 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.