Overview

The arsenal of statistical distributions is truly inexhaustible. New distributions are being discovered literally on a weekly basis prompted by either theoretical considerations or by pressing practical applications or both. Glance through any recent issue of "Annals of Statistics" or "Statistics in Medicine" to be reassured by the validity of this assertion. Each of these distributions can, of course, be used for studying the probability P(X and its estimation and we hope that the readers will engage themselves in this rewarding activity. We shall refer to this type of investigations as the "mainstream" case of the stress-strength models. These situations were tackled in the preceeding chapters involving most popular and interesting-in our subjective opinion-distributions. We also studied the case when X and Y are random vectors and the probabilities of inequalities of some linear combination of X and Y are estimated. These two models, however, do not cover all the situations that appear in various applications or theoretical exercises.

The first group of additional models studied in this chapter stems from the problems in system reliability and naturally leads to estimation of probabilities of inequalities of the type X<max( Y ₁, , Y _k) and their modifications. These models will be considered in Section 6.1. Estimation of probabilities of inequalities other than X< Y which are not covered by the previous sections (e.g. P( X) or P( X _{1 2}< < X _m)) will constitute the...