Bayesian estimation for the log-Fréchet regression model with censored lifetime data

Abstract

This paper develops a Bayesian estimation framework for the logarithmic Fréchet regression model with right-censored lifetime data. We establish four theoretical results: model identifiability via the Gumbel characteristic function, posterior propriety under normalhalf-normal priors, validity of the Bayesian predictive survival function, and an explicit deviance information criterion formula for model comparison. A Metropolis-Hastings Markov chain Monte Carlo algorithm is implemented, and Bayes estimators are derived under both squared error and linear-exponential loss, together with highest posterior density credible intervals. A Monte Carlo simulation study compares the maximum likelihood and Bayesian estimators across sample sizes and censoring rates. The model is applied to four real censored datasets, Veterans’ lung cancer, tongue cancer, Stanford heart transplant data, and annual peak river discharge. Then, it is compared against log-Weibull and log-logistic competitors, matched for parameter count within each data set, using Akaike information criteria and deviance information criterion for the three medical datasets, and Akaike information criterion and Watanabe-Akaike information criterion for the fourth. Across all applications, the log-logistic model provides the best overall fit; the log-Fréchet model outperforms the log-Weibull model most decisively on flood-discharge data, consistent with its mechanistic suitability for maximum-type generating processes rather than heavy-tailedness alone. The approach extends directly to other log-location-scale models.