Bayesian Subset Selection for Weibull Populations

Selecting the best population is an important problem in reliability analysis, particularly when lifetime data are right censored and the populations are characterized by threshold behavior. This paper presents a Bayesian subset selection procedure for multiple populations modeled by three-parameter Weibull distributions. The best population is defined as the population having the largest threshold parameter, which represents the minimum lifetime before failure can occur. Posterior probabilities of being the best are estimated based on Markov Chain Monte Carlo samples, and the selected subset is defined as the smallest collection of populations whose cumulative posterior probability of containing the best population reaches a prespecified level. Posterior inference is conducted using a random-walk Metropolis algorithm with a transformed parameterization that maintains the constraints of the three-parameter Weibull model. Convergence and sampling efficiency are evaluated through graphical and numerical diagnostics. The methodology is demonstrated using a semi-simulated fatigue-life study based on right-censored Alloy T7987 data. Model adequacy is assessed with posterior predictive methods, and posterior estimates are reported for the Weibull threshold, shape, and scale parameters for each population. In the Alloy-based study, the posterior probability of being best was distributed primarily among three populations, reflecting substantial uncertainty in the population ranking. At a target posterior probability level of 0.95, these three populations constituted the selected subset. The findings demonstrate that the proposed procedure incorporates posterior uncertainty and prevents the forced selection of a single population when the lifetime data do not offer clear separation. This approach offers a flexible Bayesian framework for selection among right-censored Weibull reliability populations.