基于广义逐次截尾数据的逆Weibull分布可靠性推断
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摘要
可靠性统计是对产品进行可靠性分析和设计的重要环节.基于许多原因,例如为了节约试验时间,或者为了降低试验成本,或者是由于试验技术的不成熟,逐次截尾试验数据是进行产品可靠性分析时经常面临的数据.所谓逐次截尾试验,是指产品试验进行到一定时刻,从正在进行试验的产品中随机的移走部分产品,余下的产品继续进行试验,直到试验结束.众所周知,在有适当先验的条件下,特别是对小样本问题,Bayes统计往往优于经典统计.而逆Weibull分布是广泛应用于力学,金融,医学等可靠性分析中的一个具有倒浴盆状失效率的统计模型.本文的主要内容就是在广义逐次截尾的试验数据下对逆Weibull分布应用Bayes统计方法进行可靠性推断,并把它推广到’Weibull分布以及另外一个修正的Weubull分布情形.同时提出一个新的修正的Weibull分布,分析其统计特性,并用实际说明其潜在的应用价值.
本文首先给出广义逐次定时截尾数据的概念以及其似然函数.接下来讨论逆Weibull分布的参数,可靠度和失效率的极大似然估计.给出基于观测信息矩阵和Bootstrap方法构造的参数的置信区间.在假设尺度参数具有gamma先验而形状参数具有对数上凸密度先验的条件下证明逆Weibull分布参数的满条件后验密度均为对数上凸函数.于是提出用Gibbs抽样方法获得Markov Chain Monte Carlo(记为MCMC)样本,由此获得参数,可靠度和失效率的Bayes估计以及双样本预测方法.总结常见的逐次截尾试验数据(统称为广义逐次截尾试验数据),分析发现Gibbs方法对广义逐次截尾试验数据仍是成立的.通过随机模拟,对逆Weibull分布参数的极大似然估计以及由观测信息矩阵构造的置信区间与其Bayes参数估计与可信区间进行比较.通过对一个服从逆Weibull分布的实例进行模拟,比较参数,可靠度和失效率的估计和Bootstrap方法构造的置信区间与其Bayes估计的异同.模拟的结果说明在有适当的先验之下,Bayes估计优于极大似然估计,而且Bayes方法对未知样本的预测很方便.进一步地,将提出的Gibbs方法推广到基于广义逐次截尾试验数据下的Weibull分布以及一个新近提出的具有修正的浴盆状失效率的分布,并用一个实例对此新模型模拟之.因此说,本文提出的Gibbs方法具有普遍性.
Weibull分布是模拟具有单调失效率数据的常用分布.但是其不能用来模拟具有(倒)浴盆状失效率的可靠性模型.于是近年来有许多修正的Weibull分布被提出来用以修正Weibull分布只有单调失效率的弱点.本文引进一个具有三参数的新的修正的Weibull分布,证明其具有单调增加和倒浴盆状的失效率,给出失效率变点的解析解.证明基于广义逐次截尾试验数据下Gibbs抽样方法仍可用于获得此模型的Bayes估计.对一个实例,通过比较新模型与已知的Weibull模型,逆Weibull模型,混合Weibull模型以及Marshall-Olkin extended Weibull模型的赤池信息准则,Bayes信息准则和修正的赤池信息准则值说明新模型潜在的应用价值.对此例的随机模拟指出对于密度不对称的分布而言,Bayes方法的区间估计优于其极大似然估计方法.
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.
本文首先给出广义逐次定时截尾数据的概念以及其似然函数.接下来讨论逆Weibull分布的参数,可靠度和失效率的极大似然估计.给出基于观测信息矩阵和Bootstrap方法构造的参数的置信区间.在假设尺度参数具有gamma先验而形状参数具有对数上凸密度先验的条件下证明逆Weibull分布参数的满条件后验密度均为对数上凸函数.于是提出用Gibbs抽样方法获得Markov Chain Monte Carlo(记为MCMC)样本,由此获得参数,可靠度和失效率的Bayes估计以及双样本预测方法.总结常见的逐次截尾试验数据(统称为广义逐次截尾试验数据),分析发现Gibbs方法对广义逐次截尾试验数据仍是成立的.通过随机模拟,对逆Weibull分布参数的极大似然估计以及由观测信息矩阵构造的置信区间与其Bayes参数估计与可信区间进行比较.通过对一个服从逆Weibull分布的实例进行模拟,比较参数,可靠度和失效率的估计和Bootstrap方法构造的置信区间与其Bayes估计的异同.模拟的结果说明在有适当的先验之下,Bayes估计优于极大似然估计,而且Bayes方法对未知样本的预测很方便.进一步地,将提出的Gibbs方法推广到基于广义逐次截尾试验数据下的Weibull分布以及一个新近提出的具有修正的浴盆状失效率的分布,并用一个实例对此新模型模拟之.因此说,本文提出的Gibbs方法具有普遍性.
Weibull分布是模拟具有单调失效率数据的常用分布.但是其不能用来模拟具有(倒)浴盆状失效率的可靠性模型.于是近年来有许多修正的Weibull分布被提出来用以修正Weibull分布只有单调失效率的弱点.本文引进一个具有三参数的新的修正的Weibull分布,证明其具有单调增加和倒浴盆状的失效率,给出失效率变点的解析解.证明基于广义逐次截尾试验数据下Gibbs抽样方法仍可用于获得此模型的Bayes估计.对一个实例,通过比较新模型与已知的Weibull模型,逆Weibull模型,混合Weibull模型以及Marshall-Olkin extended Weibull模型的赤池信息准则,Bayes信息准则和修正的赤池信息准则值说明新模型潜在的应用价值.对此例的随机模拟指出对于密度不对称的分布而言,Bayes方法的区间估计优于其极大似然估计方法.
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.
引文
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