A copula-based approach to account for dependence in stress-strength models

The focus of stress-strength models is on the evaluation of the probability R=P(Y < X) that stress Y experienced by a component does not exceed strength X required to overcome it. In reliability studies, X and Y are typically modeled as independent. Nevertheless, in many applications such an assumption may be unrealistic. This is an interesting methodological issue, especially as the estimation of R for dependent stress and strength has received only limited attention to date. This paper aims to fill this gap by evaluating R taking into account the association between X and Y via a copula-based approach. We calculate a closed-form expression for R by modeling the dependence through a Farlie-Gumbel-Morgenstern copula and one of its extensions, numerical solutions for R are, instead, provided when members of Frank's copula family are employed. The marginal distributions are assumed to belong to the Burr system (i.e.\ Burr III, Dagum or Singh-Maddala type). In all the cases, we prove that neglect of the existing dependence leads to higher or lower values of R than is the case.