A Robust ABC-Gibbs Method for Bathtub Failure Rate Modeling: Theory, Simulation, and Reliability Assessment

Reliability data in engineering and biomedical studies often exhibit non-monotone failure mechanisms, including bathtub-shaped failure rates (BTFRs). Existing flexible lifetime models capture such behavior, but statistical inference is challenging due to intractable or unstable likelihoods. We propose a likelihood-free Bayesian framework for BTFR models using approximate Bayesian computation Gibbs (ABC-Gibbs) methodology. To support this, a new competing-risk lifetime model is introduced by combining an extended Weibull and a Chen distribution in series, capable of modeling diverse non-monotone patterns, including flat bathtub regions. Within ABC-Gibbs, parameters are estimated without explicit likelihood evaluation using the Kolmogorov–Smirnov statistic. Key reliability characteristics—failure rate, mean residual life, and their change points—are investigated, providing insights for burn-in optimization and system aging. Model features such as cumulative residual entropy and Kullback–Leibler divergence are analyzed. Simulations and real data sets demonstrate that the ABC-Gibbs method is a robust and effective alternative to existing BTFR models for complex reliability analysis.