Reliability Analyses For BS Distribution

The life of products has been received the widespread attention.BS(Birnaum-Saunders)distribution is an ingenious probabilistic model that describes fatigue life.Therefore,BS dis-tribution is very suitable for describing the fatigue product lifetimes and it has been widely used in reliability.The research of reliability of two products' life is the core problem in statistical inference.Besides,the censored sample has attracted much attention in prac-tice because of time,human resources,financial resources and so on.In this paper,under the assumption X and Y are independent BS distributions,the Bayes reliability estimation R = P(Y<X)and un-observed data are analyzed when samples are progressive Type-II censoring.The background of the research,the domestic and abroad research status about this pa-per are introduced in first chapter.The character of BS fatigue life distribution,progressive Type-II censoring scheme and general purpose approximate Goodness-of-Fit test for the B-S distribution based on progressively Type-II censored samples are summarized in second chapter.In third chapter,the modified Cram6r-von Mises statistics for the null hypothesis of the BS model based on progressively Type-II censored samples are simulated.Then,the Bayes estimation of BS parameter and R = P(Y<X)are derived based on Markov Chain Monte Carlo(MCMC)technique under different loss functions duo to the Bayes estimate cannot be derived in closed form.Finally,two real data sets are presented for illustrative purposes.The examples show that the BS parameters Bayes estimates,the reliability R and corresponding standard mean square are very close under the different loss functions including the square loss,the generalized entropy loss and the absolute value loss.In fourth chapter,firstly,Bayesian point and corresponding credible interval for the un-observed data are predicted by means of MCMC under different loss functions.Secondly,the maximum likelihood estimation(MLE)of the un-observed data and corresponding confi-dence intervals are predicted based on predictive likelihood function.Finally,two examples,i.e.Examples 4.1 and 4.2,are illustrated based on different prediction methods to illustrate the above theory and method.The simulations from Examples 4.1 and 4.2 shown that the prediction values arising from Bayesian and MLE are very close.Among them,the predic-tion values based on square loss are maximum,and maximum likelihood prediction values are minimum.In addition,Bayesian prediction and condition median prediction intervals'length of Xj:rk(Yj:r'k)are increasing when eensored order is from 1 to rk(r'k).At the giv-en credible(confidence)level,the prediction intervals' lengths are very similar in terms of Bayesian and MLE prediction.