Open Access

ARTICLE

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

Osnamir E. Bru-Cordero¹, Cecilia Castro², Víctor Leiva^3,*, Mario C. Jaramillo-Elorza⁴

1 Dirección Académica, Universidad Nacional de Colombia, Sede De La Paz, La Paz, Cesar, 202010, Colombia
2 Centre of Mathematics, Universidade do Minho, Braga, 4710-057, Portugal
3 Escuela de Ingeniería Industrial, Pontificia Universidad Católica de Valparaíso, Valparaíso, 2362807, Chile
4 Departamento de Estadística, Universidad Nacional de Colombia, Sede Medellín, Medellín, 050023, Colombia

* Corresponding Author: Víctor Leiva. Email:

Computer Modeling in Engineering & Sciences 2025, 143(2), 1401-1431. https://doi.org/10.32604/cmes.2025.064487

Received 17 February 2025; Accepted 06 May 2025; Issue published 30 May 2025

Abstract

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.

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