Tomasz Rychlik - Böcker

This monograph presents a general method of establishing explicit solutions to classical problems of calculating the best lower and upper mean-variance bounds on various statistical functionals over various nonparametric families of distributions. The functionals include quantiles, standard and conditional expectations of record and order statistics from independent and dependent samples, and a variety of their combinations important in statistics and reliability. The following families of distributions are taken into account: arbitrary, symmetric, symmetric unimodal, and U-shaped ones, distributions with monotone density and failure rate, and monotone density and failure rate on the average distributions. The method is based on determining projections of the functionals onto properly chosen convex cones in functional Hilbert spaces. It allows us to explicitly point out the distributions which attain the bounds. The book is addressed to students, researchers, and practitioners in statistics and applied probability. Most of the results have been established recently, and a significant part of them has not been published yet. Numerous open problems are stated in the text.

The book discusses various inequalities and sharp bounds for the usual order statistics as well as some functions of them.

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.