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 F, attention is focused on notions of bivariate aging that can be described in terms of properties of the level curves of F. 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 copulae 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 F, 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. (c) 2004 Published by Elsevier Inc

Semi-copulas and Interpretations of Coincidences Between Stochastic Dependence and Ageing

“…B F has been also called bivariate ageing function; we also refer to it (generally, for the multivariate case) with the term B-function. For our purposes it is useful to notice here that it is Archimedean if and only if K F is such; for other details about B F see [12,21,9,10,11]. Actually, however, it turns out that B F is not generally a copula.…”

“…For this reason we can use the above equivalences (a) -(d) in order to formally define properties of positive dependence for functions of the form C φ , even when φ is not convex. If φ is not convex, then C φ is not a copula; actually it is an Archimedean semicopula or a t-norm (see [12,27]. Qualitative properties such as PQD, PKD, LTD, SI cannot anymore be properly interpreted as properties of stochastic dependence between two random variables.…”

“…Some more aspects of duality between ageing and dependence also emerge in the papers [12,24]. [12] concerns with properties of bivariate ageing and their relations with univariate ageing and dependence.…”

On dynamic mutual information for bivariate lifetimes

“…In this section we introduce the mutual information for the bivariate past lifetimes defined in (4). To this aim we consider the marginal past lifetimes…”

“…We recall that a pair of random lifetimes (X, Y ) is said to follow the time-transformed exponential (TTE) model if its joint survival function may be expressed in the following way: [4], [22], [24], [28], and [31].…”