Estimation of P[Y< X] for Dependence of Stress–Strength Models with Weibull Marginals

The stress–strength model is a basic tool used in evaluating the reliability R= P(Y< X) . We consider an expression for R where the random variables X and Y denote strength and stress, respectively. The system fails only if the stress exceeds the strength. We aim to study the effect of the dependency between X and Y on R. We assume that X and Y follow Weibull distributions and their dependency is modeled by a copula with the dependency parameter θ . We compute R for Farlie–Gumbel–Morgenstern (FGM), Ali–Mikhail–Haq (AMH), Gumbel’s bivariate exponential copulas, and for Gumbel–Hougaard (GH) copula using a Monte-Carlo integration technique. We plot the graph of R versus θ to study the effect of dependency on R. We estimate R by plugging in the estimates of the marginal parameters and of θ in its expression. The estimates of the marginal parameters are based on the marginal likelihood. The estimates of θ are obtained from two different methods; one is based on the conditional likelihood and the other is based on the method of moments using Blomqvist’s beta. Asymptotic distribution of both the estimators of R is obtained. Finally, analysis of real data set is also performed for illustrative purposes. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.