Bayesian and non-Bayesian inferences of the Burr-XII distribution for progressive first-failure censored data

In this paper, based on a new type of censoring scheme called a progressive first-failure censored, the maximum likelihood (ML) and the Bayes estimators for the two unknown parameters of the Burr type XII distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. When the two parameters are unknown, the Bayes estimators can not be obtained in explicit forms. We use Lindley's approximation to compute the Bayes estimates and the Gibbs sampling procedure to calculate the credible intervals. A Bayesian approach using Markov Chain Monte Carlo (MCMC) techniques to generate from the posterior distributions and in turn computing the Bayes estimators is developed. Point estimation and confidence intervals based on maximum likelihood and bootstrap methods are also proposed. The approximate Bayes estimators have been obtained under the assumptions of informative and non-informative priors. A numerical example using real data set is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study.