Bayesian modeling of bathtub shaped hazard rate using various Weibull extensions and related issues of model selection

Bathtub shape is one of the most important behaviors of the hazard rate function that is quite common in lifetime data analysis. Such shapes are actually the combination of three different shapes and, as such, there have been several proposals to model such behavior. One such proposal is to combine at most three different distributions, often the Weibull or some similar model, separately for decreasing, constant, and increasing shapes of the hazard rate. Sometimes combination of two different models may also result in the required bathtub shape. The other proposal includes generalizing or modifying the two-parameter distribution by adding an extra parameter into it. It is often seen that the first proposal is quite cumbersome whereas the second fails to capture some important aspects of the data. The present work considers two recent generalizations/modifications of the two-parameter Weibull model, namely the Weibull extension and the modified Weibull models, and proposes mixing the two families separately with the three-parameter Weibull distribution in order to see if the mixing results in some real benefit though at the cost of too many parameters. The paper finally considers the complete Bayes analysis of the proposed models using Markov chain Monte Carlo simulation and compares them with both Weibull extension and the modified Weibull models in a Bayesian framework. It is observed that the mixture models offer drastic improvement over the individual models not only in terms of hazard rate but also in terms of overall performance. The results are illustrated with the help of a real data based example.