Reliability Inference For The Inverse Weibull Model Based On General Progressive Censoring

Reliability statistics have a great importance in the reliability analysis anddesign for products. The demanding of time and cost reduction as well as thelimitation of technologies, progressive censoring life test is prefered in reliabil-ity engineering. Progressively censored samples are observed when, at variousstages of an experiment, some of the surviving units are removed from furtherobservation. The remainings are then continued on test under observation, ei-ther until failure, or until a subsequent stage of censoring. It is well knownthat with appropriate prior information, Bayesian inference is superior to theclassical inference for the reliability analysis, particularly for the small samplesize analysis. The inverse Weibull distribution is a products lifetime probabilitydistribution with upside-down bathtub shaped failure rate which can be used inthe reliability engineering, pharmacy and other aspects. Based on general pro-gressive censored samples, this paper proposed the Bayesian method to obtainthe interence of the inverse Weibull distrubution by using the Markov ChainMonte Carlo (MCMC) process and then it is extened to the Weibull model anda modified Weibull model.In this paper, the general progressive type-I censored data and the cor-responding likelihood function are primarily introduced. Then, the maximumlikelihood method and the Bayesian method are proposed to obtain the param-eter estimates, as well as the reliability and failure rate of the inverse Weibulldistribution based on general progressive type-I censored data. The observedFisher matrix and the Bootstrap methods are applied to construct the confi-dence interval. According to the assumption that the prior on scale parameteris the gamma density function and prior on shape parameter is the log-concavedensity function, the posterior densities of parameters are proved to be thelog-concave density function. The Gibbs sampling procedure is used to drawthe MCMC samples, and then be used to compute the Bayesian estimates as well as to construct the corresponding credible intervals of the inverse Weibulldistribution. Two-sample Bayesian prediction problem is proposed to providethe intervals of unobserved samples. The random simulation shown that theproposed Bayesian parameter estimation and the credible by using the Gibbssampling are superior to the maximum likelihood estimates and the confidenceby using the information matrix if model has the appropriate prior information.Otherwise, both results are the same. One real data analysis is performed toillustrate the application in practice. The Gibbs sampling procedure is thenextended to general progressive censored scheme and further extended to theWeibull distribution and the flexible Weibull distrbution which has a modifiedbathtub shaped failure rate. A real lifetime data set is also used to illustratethe extended results for the flexible Weibull distrbution. It is shown that theproposed Gibbs sampling method has universality. The Weibull distribution isa popular distribution for modeling phenomenon with monotonic failure rates.However, this distribution does not provide a good fit to data sets with bathtubshaped or upside-down bathtub shaped failure rates which often encountered inreliability and engineering studies.This paper introduced a new extended Weibull distribution with three pa-rameters and studied its properties. It has been found that the failure rateof the new model has increasing and upside-down bathtub shaped failure ratefunction. Based on general progressive censored data, the maximum likelihoodand Bayesian approaches are presented to estimate the unknown parameters ofthe new model. Studies indicated that the Gibbs sampling technique proposedfor the inverse Weibull distribution can be also used to construct the estimatesof the new model under the assumption that the prior on scale parameter isthe gamma density and priors on shape parameter is the log-concave densityfunction. A real data set is analyzed for illustrating the applications of the newmodel by comparing values of Akaike Information Criterion, Bayesian Informa-tion Criterion and the second order Akaike information criterion to the Weibull, inverse Weibull, mixed Weibull and Marshall-Olkin extended Weibull models,and further be used to illustrate the Bayesian method and point out that theBayesian method is superior to the maximum likelihood method when densityof the parameter is asymmetric.