A comparison of life distributions in Bayesian reliability theory

Abstract

Reliability analysis plays a fundamental role in assessing the lifetime behaviour of components, systems, and materials. In the Bayesian approach, the uncertainty about model parameters can be measured using the posterior distribution. This study presents a comparative analysis of two life distributions, the Weibull and Birnbaum-Saunders distributions under Bayesian reliability theory. The study focuses on the derivation of posterior distributions using a range of objective priors, including the Jeffreys prior, divergence prior, reference prior, and the probability matching prior, for both complete and type I right censoring cases. These priors are derived from the Fisher information matrix for both models, and the properness of the resulting posterior distributions is examined both graphically and analytically. Markov Chain Monte Carlo techniques, including the Metropolis-Hastings sampler, Gibbs sampler, and the Metropolis-within-Gibbs algorithm, are employed to simulate from the posterior distributions of the model parameters. Convergence of the posterior samples is assessed using standard diagnostics such as the trace plots, the Gelman-Rubin convergence diagnostic, and the Geweke diagnostic. Simulation studies are conducted to assess model performance across different sample sizes and priors, with evaluation based on coverage rates and mean interval lengths. Predictive reliability analyses are performed to analyse the ability of both distributions to predict future lifetimes. Applications include fitting and evaluating two fatigue lifetime datasets using both the Weibull and the Birnbaum-Saunders distributions. Bayesian estimation is carried out, and posterior summaries are analysed to assess parameter behaviour, credible intervals, and overall model fit. Model comparison using the deviance information criterion (DIC) is performed to determine which distribution provides a better fit and more stable parameter estimates.