Bayesian estimation of Weibull mixture in heavily censored data setting

In reliability or warranty analysis, engineers must often deal with lifetimes data that are non-homogeneous. Most of the time, this variability is unobserved but has to be taken into account for reliability or warranty cost analysis. A further problem is that in reliability analysis, data are heavily censored which makes estimations more difficult. The two-component Weibull mixture is then a highly relevant model to capture heterogeneity for a large majority of operating lifetimes. Unfortunately, the performance of classical estimation methods (maximum of likelihood via EM, Bayes approach via MCMC) is jeopardized due to the high number of parameters and the heavy censoring. In order to overcome the problem of heavy censoring for Weibull mixture parameters estimation, this research proposes a Bayesian bootstrap method, called Bayesian Restoration Maximization. The key is to provide a sampling from the posterior distribution. Thanks to an importance sampling technique, this sample focuses on the posterior mean. Prior distributions elicitation and sensibility analysis are discussed. Simulations results showed that, for heavily censored data, the BRM method outperforms the EM and S-EM algorithms in terms of estimates accuracy. On the other hand, it is a non-iterative method which therefore provides very short computation times. In addition, the BRM method does not suffer from the problem of label switching from Bayesian sampling algorithms such as MCMC methods. Finally, two real data sets are analyzed to illustrate the application of the method.