Conditions on marginals and copula of component lifetimes for signature representation of system lifetime

Abstract

The signature of a system is a probability vector that depends only on the system structure. Under the classic IID (independent and identically distributed) assumption on the component lifetimes, the system lifetime distribution is the convex combination of consecutive component failure times, and the signature coordinates constitute the mixture coefficients. In this case the signature representations are very useful in determining the system lifetime distributions and for stochastic comparisons of them. This first representation was obtained in 1985 by Samaniego. Then it was extended to the more general case of exchangeable component lifetimes. In 2011 Marichal, Mathonet and Waldhauser presented necessary and sufficient conditions assuring the Samaniego representation. There were expressed in terms of distributional properties of families of auxiliary indicator random vectors parametrized by positive numbers. In the paper we obtain other necessary and sufficient conditions represented in terms of the marginal distributions of component lifetimes and the dependence copula of them. Moreover, we study symmetry conditions for the equality of structural and probabilistic signatures.