Bayesian Analysis of Shared Frailty Models for Repairable Systems Subject to Imperfect Repair

Repairable systems, crucial in reliability studies, are characterized by recurrent failure times modeled as counting processes with intensity functions. This paper explores models for these failure times incorporating imperfect repairs, addressing unobserved heterogeneity via shared frailty models. In this context, our approach involves scenarios with general imperfect repairs, which offer a more realistic perspective compared to the minimal or perfect repair assumptions commonly employed in the reliability literature. We propose hierarchical Bayesian methods to estimate parameters, leveraging the Power-Law Process for initial intensities and gamma distributions for frailty terms. Bayesian methods are highly flexible and can accommodate complex shared frailty models that include random effects and dependencies between units. Applying Bayesian inference with gamma and beta distribution priors, coupled with Monte Carlo simulations, provides a robust methodology for estimating unknown parameters and deriving posterior distributions. This flexibility is crucial for capturing the underlying structure of the data in repairable systems with imperfect repairs. Our hierarchical Bayesian framework accommodates multiple systems, providing insights into failure processes and supporting enhanced maintenance strategies. We demonstrate our approach using a real failure times dataset and evaluate its performance through simulation studies, showcasing its applicability and relevance in practical settings.