Bayesian estimation based on progressive Type-II censoring from two-parameter bathtub-shaped lifetime model: an Markov chain Monte Carlo approach

In this paper, maximum likelihood and Bayes estimators of the parameters, reliability and hazard functions have been obtained for two-parameter bathtub-shaped lifetime distribution when sample is available from progressive Type-II censoring scheme. The Markov chain Monte Carlo (MCMC) method is used to compute the Bayes estimates of the model parameters. It has been assumed that the parameters have gamma priors and they are independently distributed. Gibbs within the Metropolis-Hasting algorithm has been applied to generate MCMC samples from the posterior density function. Based on the generated samples, the Bayes estimates and highest posterior density credible intervals of the unknown parameters as well as reliability and hazard functions have been computed. The results of Bayes estimators are obtained under both the balanced-squared error loss and balanced linear-exponential (BLINEX) loss. Moreover, based on the asymptotic normality of the maximum likelihood estimators the approximate confidence intervals (CIs) are obtained. In order to construct the asymptotic CI of the reliability and hazard functions, we need to find the variance of them, which are approximated by delta and Bootstrap methods. Two real data sets have been analyzed to demonstrate how the proposed methods can be used in practice.