Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2013, Article ID 146140, 16 pages http://dx.doi.org/10.1155/2013/146140 Research Article Bayesian Estimation and Prediction for Flexible Weibull Model under Type-II Censoring Scheme Sanjay Kumar Singh, Umesh Singh, and Vikas Kumar Sharma Department of Statistics and DST-CIMS, Banaras Hindu University, Varanasi 221005, India Correspondence should be addressed to Vikas Kumar Sharma; vikasstats@redifmail.com Received 4 April 2013; Revised 3 June 2013; Accepted 18 June 2013 Academic Editor: Shein-chung Chow Copyright © 2013 Sanjay Kumar Singh et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We have developed the Bayesian estimation procedure for fexible Weibull distribution under Type-II censoring scheme assuming Jefrey’s scale invariant (noninformative) and Gamma (informative) priors for the model parameters. Te interval estimation for the model parameters has been performed through normal approximation, bootstrap, and highest posterior density (HPD) procedures. Further, we have also derived the predictive posteriors and the corresponding predictive survival functions for the future observations based on Type-II censored data from the fexible Weibull distribution. Since the predictive posteriors are not in the closed form, we proposed to use the Monte Carlo Markov chain (MCMC) methods to approximate the posteriors of interest. Te performance of the Bayes estimators has also been compared with the classical estimators of the model parameters through the Monte Carlo simulation study. A real data set representing the time between failures of secondary reactor pumps has been analysed for illustration purpose. 1. Introduction In reliability/survival analysis, generally, life test experiments are performed to check the life expectancy of the manufac- tured product or items/units before products produced in the market. But in practice, the experimenters are not able to observe the failure times of all the units placed on a life test due to time and cost constraints or due to some other uncertain reasons. Data obtained from such experiments are called censored sample. Keeping time and cost constraints in mind, many types of censoring schemes have been discussed in the statistical literature named as Type-I censoring, Type- II censoring and progressive censoring schemes, and so forth. In this paper, Type-II censoring scheme is considered. In Type-II censoring scheme, the life test is terminated as soon as a prespecifed number (say, ) of units have failed. Terefore, out of  units put on test, only frst  failures will be observed. Te data obtained from such a restrained life test will be referred to as a Type-II censored sample. Prediction of the lifetimes of future items based on censored data is very interesting and valuable topic for researchers, engineers and reliability practitioners. In predic- tive inference, In predictive Inference, we can infer about the lifetimes of the future items using observed data. Te future prediction problem can be classifed into two types: (1) one-sample prediction problem. (2) two-sample prediction problem. In one-sample prediction problem, the variable to be predicted comes from the same sequence of variables observed and dependent of the informative sample. In the second type, the variable to be predicted comes from another independent future sample. Reference [1] has developed the Bayesian procedure to the prediction problems of future observations and use the concept of Bayesian predictive posterior distribution. Many authors have focussed on the problem of Bayesian prediction of future observations based on various types of censored data from diferent lifetime models (see [2–9], and references cited therein). Te fexible Weibull distribution is a new two-parameter generalization of the Weibull model which has been proposed