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

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SUMMARYThe paper is devoted to study stochastic comparisons of series and parallel systems with vectors of component lifetimes sharing the same copula. We show that, under some conditions on the common copula, the series system with heterogeneous components is worse than the series system with homogeneous components having a common reliability function, which is equal to the average of the reliability functions of the heterogeneous components. However, we show that this property is not necessarily true for arbitrary copulas. We obtain similar properties for parallel systems and for general coherent systems. For these purposes, we introduce in our analysis the notion of the mean function of a copula.

The signature is an important structural characteristic of a coherent system. Its computation, however, is often rather involved and complex. We analyze several cases where this complexity can be considerably reduced. These are the cases when a ‘large’ coherent system is obtained as a series, parallel, or recurrent structure built from ‘small’ modules with known signature. Corresponding formulae can be obtained in terms of cumulative notions of signatures. An algebraic closure property of families of homogeneous polynomials plays a substantial role in our derivations.

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