Reliability Assessment of Systems with Bivariate Dependent Stresses: A Comparative Study of Marshall-Olkin Distribution and FGM Copula with Bayesian Inference

Hakamipour, Nooshin

doi:10.22034/IJRRS.2026.9.1.12

Reliability Assessment of Systems with Bivariate Dependent Stresses: A Comparative Study of Marshall-Olkin Distribution and FGM Copula with Bayesian Inference

Document Type : Original Research Article

Author

Nooshin Hakamipour

Department of Mathematics, Buein Zahra Technical University, Buein Zahra, Qazvin, Iran

10.22034/IJRRS.2026.9.1.12

Abstract

This paper develops a framework for the reliability of a stress-strength model that assumes the strength variable (Y) is constrained by two dependent stresses (X_1 and X_2). In contrast with classical forms of this study, which typically assume independence of the stress variables, this one considers the realistic aspect of the dependence relationship between the two variables that, if ignored, can result in biased estimates of reliability. There are two dependence structures that we use. The Marshall–Olkin model is a formulation that includes common shock effects, and the Farlie–Gumbel–Morgenstern (FGM) copula provides analytical tractability and flexible dependence models without using the marginal distributions. These two structures are compared in this study. Both maximum likelihood estimation and Bayesian estimation are employed to acquire point and interval estimates for the model parameters. The Bayesian approach is implemented with the Metropolis–Hastings algorithm. The results of the simulation indicate that the Bayesian method tends to give more precise estimates with less bias, lower Mean Squared Error, and more efficient interval estimates, particularly when the sample size is small. Both models are adequate for describing the underlying dependence; however, the FGM copula provides more accurate results for moderate dependence systems with more efficient interval estimates. The practical utility and soundness of the suggested methodologies are also demonstrated through a real-data example. In general, the paper provides a comprehensive comparison under a unified estimation framework that analyzes dual dependence structures in the estimation of R=P(X_1<Y<X_2 ) and emphasizes the critical importance of valid dependence modeling in stress-strength reliability.

Keywords

Bayesian estimation

Bivariate Stress– strength reliability

Dependent stress

FGM Copula

Marshall– Olkin distribution

Metropolis– Hastings algorithm

Subjects

Reliability Engineering

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