Approximating the Lévy-Frailty Marshall-Olkin Model for Failure Times

Abstract

In this paper we approximate the last, close-to-first, and what we call quantile failure times of a system, when the system-components’ failure times are modeled according to a Levy-frailty Marshall-Olkin (LFMO) distribution. The LFMO distribution is a fairly recent model that can be used to model components failing simultaneously in groups. One of its prominent features is that the failure times of the components are conditionally iid; indeed, the failure times are iid exponential when conditioned on the path of a given Lévy subordinator process. We are motivated by further studying the order statistics of the LFMO distribution, as recently Barrera and Lagos (2020) showed an atypical behavior for the upper-order statistics. We are also motivated by approximating the system when it has an astronomically large number of components. We perform computational experiments that show significative variations in the convergence speeds of our approximations.