Sharp Inequalities for Ordered Random Variables in Statistics and Reliability
Description
The book discusses various inequalities and sharp bounds for the usual order statistics as well as some functions of them. In particular, deterministic bounds, bounds for the case of IID samples from general, symmetric and life distributions, IID samples from shape restricted family of distributions, and samples from finite populations are all discussed in detail. An elaborate numerical evaluation and comparison of various bounds are also presented in order to illustrate their inherent differences as well as their precision. Furthermore, their applications to inference, reliability theory and characterizations are also highlighted.
The book provides an in-depth exposure to various mathematical inequalities and bounds established historically as well as in recent years and their applications to order statistics and some important functions of them. It thus presents an up-to-date discussion of all results in this important area of mathematical and statistical research. The results described here are general in nature and therefore could be useful in other areas of Probability and Statistics as well.
Reviews / Votes
"Written by two renowned researchers in the field of stochastic processes, the present book is an excellent review on the inequalities and bounds related to the standard order statistics. . Numerous recent results due to the authors of this monograph are included. Some numerical evaluations adequately illustrate the theoretical results. The results presented in this book have important applications in reliability theory. The reading of the theorems and their proofs is particularly instructive and enjoyable." (Eugen Paltanea, zbMATH 1568.60001, 2025)
"The book is an excellent compilation on the classic and recent results on sharp bounds for the classic order statistics (from IID samples)." (Jorge Navarro, Mathematical Reviews, May, 2025)
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Persons
Professor N. Balakrishnan is a Distinguished University Professor in the Department of Mathematics and Statistics at McMaster University, Hamilton, Canada. He is a Fellow of the Royal Society of Canada, a Fellow of the American Statistical Association, a Fellow of the Institute of Mathematical Statistics, and a recipient of an honorary doctoral degree from the National and Kapodistrian University of Athens, Greece. He is an author of many well-known books and research monographs in Statistics including the well-known four encyclopedic volumes on Statistical Distributions published by John Wiley & Sons. He is currently the Editor-in-Chief of three series of Communications in Statistics as well as Mathematical Methods of Statistics.
Professor Tomasz Rychlik received the M.Sc. Degree in applied mathematics at the Faculty of Applied Mathematics and Technical Physics of the Technical University of Warsaw, Poland, in 1982. He defended his PhD thesis on robust statistics at the Institute of Mathematics, Polish Academy of Sciences in 1986. Now he is Professor of Mathematics at the Institute of Mathematics, Polish Academy of Sciences. His research interests contain probabilistic inequalities, reliability theory, and stochastic orders. He has published over 120 papers. Most of them were devoted to inequalities for ordered random variables.
Content
- Introduction and Notation.- Analytic Inequalities and Other Tools.- Deterministic Bounds.- General, Symmetric and Life IID Samples.- Sampling from Finite Populations.- IID Samples from Shape Restricted Families.