Relations among univariate aging, bivariate aging and dependence for exchangeable lifetimes

Abstract

For a couple of lifetimes ( X 1, X 2) with an exchangeable joint survival function F̄, attention is focused on notions of bivariate aging that can be described in terms of properties of the level curves of F̄. We analyze the relations existing among those notions of bivariate aging, univariate aging, and dependence. A goal and, at the same time, a method to this purpose is to define axiomatically a correspondence among those objects; in fact, we characterize notions of univariate and bivariate aging in terms of properties of dependence. Dependence between two lifetimes will be described in terms of their survival copula. The language of copulæ turns out to be generally useful for our purposes; in particular, we shall introduce the more general notion of semicopula. It will be seen that this is a natural object for our analysis. Our definitions and subsequent results will be illustrated by considering a few remarkable cases; in particular, we find some necessary or sufficient conditions for Schur-concavity of F̄, or for IFR properties of the one-dimensional marginals. The case characterized by the condition that the survival copula of ( X 1, X 2) is Archimedean will be considered in some detail. For most of our arguments, the extension to the case of n>2 is straightforward.