Tomasz Rychlik

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.

About 10 years ago I began studying evaluations of distributions of or­ der statistics from samples with general dependence structure. Analyzing in [78] deterministic inequalities for arbitrary linear combinations of order statistics expressed in terms of sample moments, I observed that we obtain the optimal bounds once we replace the vectors of original coefficients of the linear combinations by the respective Euclidean norm projections onto the convex cone of vectors with nondecreasing coordinates. I further veri­ fied that various optimal evaluations of order and record statistics, derived earlier by use of diverse techniques, may be expressed by means of projec­ tions. In Gajek and Rychlik [32], we formulated for the first time an idea of applying projections onto convex cones for determining accurate moment bounds on the expectations of order statistics. Also for the first time, we presented such evaluations for non parametric families of distributions dif­ ferent from families of arbitrary, symmetric, and nonnegative distributions. We realized that this approach makes it possible to evaluate various func­ tionals of great importance in applied probability and statistics in different restricted families of distributions. The purpose of this monograph is to present the method of using pro­ jections of elements of functional Hilbert spaces onto convex cones for es­ tablishing optimal mean-variance bounds of statistical functionals, and its wide range of applications. This is intended for students, researchers, and practitioners in probability, statistics, and reliability.