3.4 Elliptical Distributions

Elliptical (or elliptically contoured) distributions have been popular since early eighties as a natural extension of the multivariate normal distribution with attractive properties, both theoretical and applications oriented (see e.g. Fang et al. (1990)).

In this and the next section we shall investigate a generalization of the stress-strength model where instead of a two-component vector ( X, Y) we have two independent k ₁ and k ₂-component random vectors and are used. We shall be interested in estimating the probability P( A ?X+B ?Y+ C> 0) where A and B are known k ₁ and k ₂-dimensional vectors and C is a known scalar. This setting and its minor variations were considered in the case when X and Y are normally distributed random vectors by Pensky (1982), Gupta and Gupta (1990) and Ivshin and Lumelskii (1994). It is easy to see that in the one-dimensional case when k ₁= k ₂=1 , C=0, A=( ?1) and B=1, the problem reduces to estimation of the probability P(X for independent random variables X and Y. The case when k ₁=2, A=( ?1, 1) and B=0 allows us to estimate P( X< Y) for dependent variables X and Y.

Estimation of P( A ?X+B ?Y+ C>0) is very important...