On the Importance of Copula Choice in the Reliability Evaluation of Dependent Stress-Strength Models

Reliability assessment, vital in high-stakes engineering, often employs the stress-strength model. However, traditional models frequently assume independence between stress and strength, an assumption that can lead to inaccurate reliability estimates when dependence exists due to real-world factors. To address this, the current study proposes a dependent stress-strength model using copula theory, which flexibly models dependence by separating marginal and joint distributions. Four copula families—Farlie-Gumbel-Morgenstern, Ali-Mikhail-Haq, Gumbel's bivariate exponential, and Gumbel-Hougaard—are investigated for their ability to capture diverse dependency patterns. The Inverse Lomax distribution is utilized for both stress and strength marginals due to its suitability for heavy-tailed reliability data. The copula dependence parameter θ is estimated via conditional likelihood and Blomqvist's beta-based method of moments. The asymptotic distributions of these estimators are derived, and their performance is evaluated through extensive simulations. The research thoroughly examines how system reliability R changes with $\theta$ across various model configurations. Findings indicate that the Gumbel-Hougaard copula demonstrates the highest sensitivity of $R$ to θ, effectively capturing a wide range of dependency strengths. This paper highlights the critical need to incorporate dependence in stress-strength models and offers practical guidance for copula selection, thereby enhancing the accuracy and robustness of reliability predictions in complex engineering systems. A practical examination of a real dataset is conducted to demonstrate the concept.