Statistical Analysis Of Reliability For FW Distribution Based On Record Values

The two parameters flexible Weibull(FW)distribution is proposed by Bebbington et al.(2007)to model the life time.The failure rate of the FW distribution can be increased or modified bathtub with appropriate choice of parameter values.As a special case of order statistics,record value has been widely applied to the reliability analysis and other areas.In this paper,the classical statistical method and Bayesian statistical method are used to study the properties of the FW distribution.The definition,properties and likelihood function of record value are discussed in Chapter 2.In Chapter 3,the failure rate and WPP plot of the FW model are primarily analyzed.Then,the regression estimation,inverse moment estimation,maximum likelihood estimation(MLE)techniques are used to estimate the parameter,reliability and failure rate of the FW distribution.And,observation information matrix and Bootstrap methods are applied to approximate variances and confidence intervals(CIs)of the estimates.Under the assumption that the priors on the two-parameter are the gamma density functions,we proved that the full conditional posterior distribution of the two parameters both are log-concave.Thus,the Gibbs sampling technique is used to obtain the Markov chain Monte Carlo(MCMC)sample.Based on the MCMC sample,Bayesian estimations including parameter,reliability,and failure rate and the corresponding credible interval(CI)are derived.Finally,the real data are used to illustrate the proposed methods.Simulation results shown that the Bayesian estimation is better than the classical estimation based on mean square error(MSE)and length of CI.It is also pointed out that the precision from the MLE is the best,next is the regression estimation,and the regression estimation is the worst.The reliability statistics of the FW distribution are discussed in Chapter 4 based on the record values.The MLEs of parameters,reliability and failure rate is primarily discussed,and the observation information matrix is applied to construct CI.Then,it is proved that the full conditional posterior densities of the parameters are log-concave based on different priors.The Gibbs sampling procedure is used to obtain Bayesianestimations and posterior CIs of parameter,reliability and failure rate.Finally,numerical simulation shown that Bayesian estimation is superior to MLE if the model has the appropriate prior information.