Confidence Intervals for the Reliability of Dependent Systems: Integrating Frailty Models and Copula-Based Methods

Most reliability studies assume large samples or independence among components, but these assumptions often fail in practice, leading to imprecise inference. We address this issue by constructing confidence intervals (CIs) for the reliability of two-component systems with Weibull distributed failure times under a copula-frailty framework. Our construction integrates gamma-distributed frailties to capture unobserved heterogeneity and a copula-based dependence structure for correlated failures. The main contribution of this work is to derive adjusted CIs that explicitly incorporate the copula parameter in the variance-covariance matrix, achieving near-nominal coverage probabilities even in small samples or highly dependent settings. Through simulation studies, we show that, although traditional methods may suffice with moderate dependence and large samples, the proposed CIs offer notable benefits when dependence is strong or data are sparse. We further illustrate our construction with a synthetic example illustrating how penalized estimation can mitigate the issue of a degenerate Hessian matrix under high dependence and limited observations, so enabling uncertainty quantification despite deviations from nominal assumptions. Overall, our results fill a gap in reliability modeling for systems prone to correlated failures, and contribute to more robust inference in engineering, industrial, and biomedical applications.