American Journal of Theoretical and Applied Statistics 2013; 2(5): 128-141 Published online August 30, 2013 (http://www.sciencepublishinggroup.com/j/ajtas) doi: 10.11648/j.ajtas.20130205.13 Bayesian estimation using MCMC approach based on progressive first-failure censoring from generalized Pareto distribution Mohamed Abdul Wahab Mahmoud 1 , Ahmed Abo-Elmagd Soliman 2 , Ahmed Hamed Abd Ellah 3 , Rashad Mohamed El-Sagheer 1, * 1 Mathematics Department, Faculty of Science, A1-Azhar University, Nasr-City 11884, Cairo, Egypt 2 Mathematics Department, Faculty of Science, Islamic University, Madinh, Saudi Arabia 3 Mathematics Department, Faculty of Science, Sohag University, Sohag 82524, Egypt Email address: mawmahmoud11@hotmail.com (Mohamed A. W. Mahmoud), a_a_sol@hotmail.com (Ahmed A. Soliman), ahmhamed@hotmail.com (Ahmed H. Abd Ellah), Rashadmath@yahoo.com (Rashad M. El-Sagheer) To cite this article: Mohamed Abdul Wahab Mahmoud, Ahmed Abo-Elmagd Soliman, Ahmed Hamed Abd Ellah, Rashad Mohamed El-Sagheer. Bayesian Estimation Using MCMC Approach Based on Progressive First-Failure Censoring from Generalized Pareto Distribution. American Journal of Theoretical and Applied Statistics. Vol. 2, No. 5, 2013, pp. 128-141. doi: 10.11648/j.ajtas.20130205.13 Abstract: 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 Generalized Pareto (GP) distribution are derived. This type of censoring contains as special cases various types of censoring schemes used in the literature. A Bayesian approach using Markov Chain Monte Carlo (MCMC) method to generate from the posterior distributions and in turn computing the Bayes estimators are 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 is provided to illustrate the proposed methods. Finally, the maximum likelihood and different Bayes estimators are compared via a Monte Carlo simulation study. Keywords: Generalized Pareto Distribution, Progressive First-Failure Censored Sample, Gibbs and Metropolis Sampler, Bayesian and Non-Bayesian Estimations, Bootstrap Methods 1. Introduction Censoring is very common in life tests. There are several types of censored tests. One of the most common censored test is type II censoring. It is noted that one can use type II censoring for saving time and money. However, when the lifetimes of products are very high, the experimental time of a type II censoring life test can be still too long. (Johnson, 1964) described a life test in which the experimenter might decide to group the test units into several sets, each as an assembly of test units, and then run all the test units simultaneously until occurrence the first failure in each group. Such a censoring scheme is called first-failure censoring. (Jun, et.al 2006) discussed a sampling plan for a bearing manufacturer. The bearing test engineer decided to save test time by testing 50 bearings in sets of 10 each. The first-failure times from each group were observed. (Wu, et.al 2003; Wu, and Yu, 2005) obtained maximum likelihood estimates (MLEs), exact confidence intervals and exact confidence regions for the parameters of the Gompertz and Burr type XII distributions based on first-failure censored sampling, respectively. Also see (Wu, et.al 2001; Lee, et.al 2007). Recently, (Wu, and Kuş, 2009) obtained maximum likelihood estimates, exact confidence intervals and exact confidence regions for the parameters of Weibull distribution under the progressive first-failure censored sampling. Note that a first-failure censoring scheme is terminated when the first failure in each set is observed. If an experimenter desires to remove some sets of test units before observing the first failures in these sets this life test plan is called a progressive first-failure censoring scheme which recently introduced by (Wu, and Kuş, 2009). (Soliman, et.al 2012a) obtained estimation from Burr type XII distribution using progressive first-failure censored data. (Soliman, et.al 2012b) discussed estimation of the parameters of life for Gompertz distribution using progressive first-failure censored data. (Soliman, et.al 2011a) obtained Bayesian inference and prediction of Burr type XII distribution for