随机右截尾情形下威布尔分布可靠度的置信下限
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摘要
随着科学技术的发展,许多大型复杂的智能化程控设备或产品广泛应用于社会各个领域,这些设备或产品一旦发生故障,将会对社会、经济等方面造成不同程度的损害,这些损害有时甚至会危及到人民生命财产的安全。而描述这些设备或产品安全性的一个重要指标就是可靠性。因此,可靠性越来越受到人们的重视。
在产品可靠性分析、研究过程中,主要是借助统计分析方法,对产品寿命试验数据进行分析以确定其可靠性,而可靠性指标中较常用的是可靠度的置信下限。但因实验设备、观测手段、实验成本等方面的影响,造成寿命实验有时并不是同时开始,既使同时开始,也可能不同时结束。因此在实验结束时,我们就得到一组“随机右截尾数据”。
本文基于威布尔分布下的右截尾数据,对可靠度的置信下限进行讨论。首先利用极大似然估计理论对威布尔分布的参数估计进行了一般论述。然后,在此基础上采用统计量方法对威布尔分布下可靠度的置信下限的计算进行了分析。由于威布尔分布在随机右截尾下的计算比较复杂,因此利用极大似然估计的相合性和参数β的特点对参数β给出一定的限制条件,再通过威布尔分布与指数分布的转换公式,就可以比较容易的求出可靠度的近似置信下限,大大简化了计算,使其在实际中的应用成为可能。本文最后用鞍点逼近计算方法给出了可靠度的置信下限的计算公式,并通过数据模拟证明了本方法是可行的。
With the development of science and technology, a lot of large complex intelligent program control equipments or products are used in all social fields. Once these equipments or products go wrong, they will damage all social and economic aspects to some extent, and even endanger people's life-property safety. One of the import indexes of these products safety is the reliability. Therefore, it has been paid more and more attention.
In the process of analyzing and studying the reliability, it is mainly, with the aid of statistic analysis method, to analysis test data of product life in order to confirm the reliability. Moreover, the usually used index of it is the lower confidence limits of the reliability. However, being affected by experiment equipment, observation method and cost of experiment, life test does not start at the same time; even so, it may not finish simultaneously. So at the end of the experiment, we get a group of randomly censored data on the right.
Based on the right-censored data in the Weibull distribution, this paper discusses the lower confidence limits of the reliability. Above all, using maximum likelihood estimation theory, it makes a general exposition for estimating parameter of Weibull distribution. And then, on that basis using statistical estimation it analyses the calculation of the lower confidence limits of the reliability in the Weibull distribution. Because the calculation of the randomly right-censored data in the Weibull distribution is comparatively complicated, we can give certain limiting conditions to parameterβusing consistent property maximumlikelihood estimate and its own characteristic, and then relatively easily obtain the lower confidence limits of the reliability by the conversion formula of Weibull distribution and exponential distribution. This way simplifies the calculation, and can be applied actually. In the end, this paper gives the calculation formula of the lower confidence limits of the reliability by saddle point approximation algorithm, and simulative proves that this way is feasible based on data.
在产品可靠性分析、研究过程中,主要是借助统计分析方法,对产品寿命试验数据进行分析以确定其可靠性,而可靠性指标中较常用的是可靠度的置信下限。但因实验设备、观测手段、实验成本等方面的影响,造成寿命实验有时并不是同时开始,既使同时开始,也可能不同时结束。因此在实验结束时,我们就得到一组“随机右截尾数据”。
本文基于威布尔分布下的右截尾数据,对可靠度的置信下限进行讨论。首先利用极大似然估计理论对威布尔分布的参数估计进行了一般论述。然后,在此基础上采用统计量方法对威布尔分布下可靠度的置信下限的计算进行了分析。由于威布尔分布在随机右截尾下的计算比较复杂,因此利用极大似然估计的相合性和参数β的特点对参数β给出一定的限制条件,再通过威布尔分布与指数分布的转换公式,就可以比较容易的求出可靠度的近似置信下限,大大简化了计算,使其在实际中的应用成为可能。本文最后用鞍点逼近计算方法给出了可靠度的置信下限的计算公式,并通过数据模拟证明了本方法是可行的。
With the development of science and technology, a lot of large complex intelligent program control equipments or products are used in all social fields. Once these equipments or products go wrong, they will damage all social and economic aspects to some extent, and even endanger people's life-property safety. One of the import indexes of these products safety is the reliability. Therefore, it has been paid more and more attention.
In the process of analyzing and studying the reliability, it is mainly, with the aid of statistic analysis method, to analysis test data of product life in order to confirm the reliability. Moreover, the usually used index of it is the lower confidence limits of the reliability. However, being affected by experiment equipment, observation method and cost of experiment, life test does not start at the same time; even so, it may not finish simultaneously. So at the end of the experiment, we get a group of randomly censored data on the right.
Based on the right-censored data in the Weibull distribution, this paper discusses the lower confidence limits of the reliability. Above all, using maximum likelihood estimation theory, it makes a general exposition for estimating parameter of Weibull distribution. And then, on that basis using statistical estimation it analyses the calculation of the lower confidence limits of the reliability in the Weibull distribution. Because the calculation of the randomly right-censored data in the Weibull distribution is comparatively complicated, we can give certain limiting conditions to parameterβusing consistent property maximumlikelihood estimate and its own characteristic, and then relatively easily obtain the lower confidence limits of the reliability by the conversion formula of Weibull distribution and exponential distribution. This way simplifies the calculation, and can be applied actually. In the end, this paper gives the calculation formula of the lower confidence limits of the reliability by saddle point approximation algorithm, and simulative proves that this way is feasible based on data.
引文
[1]陆廷孝,郑鹏皱.可靠性设计与分析[M].长沙:国防工业出版社,1997:2-5
[2]刘贵生,胡师金,舒寅清等.如何提高机械产品的可靠性[J].洛阳工学院学报,1999.2:53-56
[3]AGREE.Reliability of Military Electronic Equipment,Report of the Advisory Group on Reliability of Electronic Equipment.Washington DC:Office of the Assistant Secretary of Defense[R],1957
[4]何波.模糊参数系统可靠性分析及应用研究[D].辽宁:辽宁工程技术大学,2006
[5]陈宁宁.机电产品可靠性管理的现状和发展趋势[J].机械制造.2006.5:56-59
[6]李同胜,郭民之.关于定时截尾无失效数据情形下Weibull分布条件可靠度的置信下限[J].云南师范大学学报(自然科学版),2006,(05):14-17
[7]蒋同斌.基于威布尔分布分布的无失效数据的可靠性分析[J].金陵科技学院学报,2005,(03):15-19
[8]Jeng S and Meeker W.Comparisons of approximates for type Ⅰ censored date[J].Technometrics,2000,42(2):135-148
[9]Chen J and Fang X.On the exact lower confidence limits for the reliability in the case of Weibull distribution under time censoring[R].Research Report(2001),School of Math Sci.Peking Univ
[10]荆广珠、房祥忠.非等定时截尾寿命试验方案指数分布情形平均寿命的置信限[J].数理统计与管理.2000,19(4):46-49
[11]严广松,侯紫燕.不相同定时截尾试验中平均寿命的置信限[J].郑州纺织工学院学报,1998,(03):63-67
[12]董岩,徐兴忠,杨雪姣.双参数指数分布的可靠寿命的广义置信下限[J].系统科学与数学,2008,(09):1109-1117
[13]胡思贵,赵明.指数分布区间型删失数据的可靠度最优置信下限[J]。贵州大学学报(自然科学版),2007,(06):571-574
[14]高峰,定数截尾数据下指数分布参数带有固定精度要求的置信下限[J].南阳师范学院学报,2004,(09):18-20
[15]赵永亮,张晓敏,张德菊等.两指数分布并联系统可靠度的置信下限[J].临沂师范学院学报,1999,(03):4-7
[16]吴和成,刘思峰.基于不确定数据指数型产品贮存可靠性的置信下限[J].中国机械工程,2006,(22):2330-2332
[17]田霆,刘次华.定数截尾场合下Weibull分布的形状参数置信下限[J].电子产品可靠性与环境试验,2007,(04):19-21
[18]庞敏,廖崇尧,朱克杰等.当m已知时威布尔(Weibull)寿命分布产品的可靠度置信下限[J].质量与可靠性,2006,(02):23-24
[19]郭维长.威布尔分布的参数估计和可靠性置信下限近似解[J].航天器工程,2006,(01):14-17
[20]于晓红,张来斌,王朝晖等.基于新的威布尔分布参数估计法的设备寿命可靠性分析[J].机械强度,2007,(06):932-936
[21]李宝盛.由威布尔分布零时效样本求任务可靠性的Fiducial-Bayes置信下限[J],质量与可靠性,2004,(05):30-32
[22]费鹤良,倪中新.威布尔分布或极值分布可靠度置信下限的回归表示[J].运筹学学报,2002,(03):77-84
[23]崔卫民,薛红军,喻天翔等.实验数据服从Weibull分布时可靠性试验最少试件数的确定[J].机械工程学报,2008,(01):51-55
[24]曹晋华,程侃著.可靠性数学引论(修订版)[M].高等教育出版社.2006:398-399
[25]陈家鼎编著,生存分析与可靠性[M].北京大学出版社.2005:141-142,221-223,227-228
[26]Wang J and Chen J.On the strong consistency of the maximum likelihood estimators from randomly censored samples[J].Inter J of Relia,Qua and Saf Engin,1997,4:35-53
[27]陈家鼎,随机截尾情形下Weibull分布参数的最大似然估计的相合性[J].应用概率统计,1989,5(3):226-233
[28]李补喜.随机删失情形分布参数的最大似然估计的重对数律[J].应用概率统计,1995,11(1):60-69
[29]陈希孺.数理统计引论[M].北京:科学出版社,1981:363-365
[30]Buehler R J.Confidence interval for the product of two Binomial parameters[J].J Amer Statist Assoc,1957,52:482-493
[31]陈家鼎.样本空间中的序与参数的置信限[J].数学进展,1993,22(6):542-552
[32]Wei zhongpeng,On the Confidence limits of the Reliability in the Case of I type Interval Censored Data[J],ACTA Mathematicae Applicatae Sinica,2006,29(1):81-90
[33]程侃著.寿命分布类与可靠性数学理论[M].北京:科学出版社,1999:494-509
[34]王瑞省.鞍点规划与模糊鞍点规划[D],河北:河北大学,2002
[35]Jens Ledet Jensen.Saddlepoint Approximations[M].New York:Oxford University Press,1995
[36]Dong Yan.The Lower Confidence limit of Reliability for Exponential life Data with Varying Censoring Time[J],ACTA Mathematicae Applicatae Sinica,2006,29(1):61-67
[37]Mao Shisong.Advanced Mathematical Statistics[M].Beijing China Higher Education Press;Berlin Heidelberg:Springer-Verlag,1998(in Chinese):271-295
[38]Lawless J F.Statistical Models and Methods for Lifetime Data[R].New York:John Wiley,1982
[39]Bartholomew D J.The Sampling Distribution of an Estimate Arising in Life Testing[J].Technometrics,1963,5(3):361-374
[2]刘贵生,胡师金,舒寅清等.如何提高机械产品的可靠性[J].洛阳工学院学报,1999.2:53-56
[3]AGREE.Reliability of Military Electronic Equipment,Report of the Advisory Group on Reliability of Electronic Equipment.Washington DC:Office of the Assistant Secretary of Defense[R],1957
[4]何波.模糊参数系统可靠性分析及应用研究[D].辽宁:辽宁工程技术大学,2006
[5]陈宁宁.机电产品可靠性管理的现状和发展趋势[J].机械制造.2006.5:56-59
[6]李同胜,郭民之.关于定时截尾无失效数据情形下Weibull分布条件可靠度的置信下限[J].云南师范大学学报(自然科学版),2006,(05):14-17
[7]蒋同斌.基于威布尔分布分布的无失效数据的可靠性分析[J].金陵科技学院学报,2005,(03):15-19
[8]Jeng S and Meeker W.Comparisons of approximates for type Ⅰ censored date[J].Technometrics,2000,42(2):135-148
[9]Chen J and Fang X.On the exact lower confidence limits for the reliability in the case of Weibull distribution under time censoring[R].Research Report(2001),School of Math Sci.Peking Univ
[10]荆广珠、房祥忠.非等定时截尾寿命试验方案指数分布情形平均寿命的置信限[J].数理统计与管理.2000,19(4):46-49
[11]严广松,侯紫燕.不相同定时截尾试验中平均寿命的置信限[J].郑州纺织工学院学报,1998,(03):63-67
[12]董岩,徐兴忠,杨雪姣.双参数指数分布的可靠寿命的广义置信下限[J].系统科学与数学,2008,(09):1109-1117
[13]胡思贵,赵明.指数分布区间型删失数据的可靠度最优置信下限[J]。贵州大学学报(自然科学版),2007,(06):571-574
[14]高峰,定数截尾数据下指数分布参数带有固定精度要求的置信下限[J].南阳师范学院学报,2004,(09):18-20
[15]赵永亮,张晓敏,张德菊等.两指数分布并联系统可靠度的置信下限[J].临沂师范学院学报,1999,(03):4-7
[16]吴和成,刘思峰.基于不确定数据指数型产品贮存可靠性的置信下限[J].中国机械工程,2006,(22):2330-2332
[17]田霆,刘次华.定数截尾场合下Weibull分布的形状参数置信下限[J].电子产品可靠性与环境试验,2007,(04):19-21
[18]庞敏,廖崇尧,朱克杰等.当m已知时威布尔(Weibull)寿命分布产品的可靠度置信下限[J].质量与可靠性,2006,(02):23-24
[19]郭维长.威布尔分布的参数估计和可靠性置信下限近似解[J].航天器工程,2006,(01):14-17
[20]于晓红,张来斌,王朝晖等.基于新的威布尔分布参数估计法的设备寿命可靠性分析[J].机械强度,2007,(06):932-936
[21]李宝盛.由威布尔分布零时效样本求任务可靠性的Fiducial-Bayes置信下限[J],质量与可靠性,2004,(05):30-32
[22]费鹤良,倪中新.威布尔分布或极值分布可靠度置信下限的回归表示[J].运筹学学报,2002,(03):77-84
[23]崔卫民,薛红军,喻天翔等.实验数据服从Weibull分布时可靠性试验最少试件数的确定[J].机械工程学报,2008,(01):51-55
[24]曹晋华,程侃著.可靠性数学引论(修订版)[M].高等教育出版社.2006:398-399
[25]陈家鼎编著,生存分析与可靠性[M].北京大学出版社.2005:141-142,221-223,227-228
[26]Wang J and Chen J.On the strong consistency of the maximum likelihood estimators from randomly censored samples[J].Inter J of Relia,Qua and Saf Engin,1997,4:35-53
[27]陈家鼎,随机截尾情形下Weibull分布参数的最大似然估计的相合性[J].应用概率统计,1989,5(3):226-233
[28]李补喜.随机删失情形分布参数的最大似然估计的重对数律[J].应用概率统计,1995,11(1):60-69
[29]陈希孺.数理统计引论[M].北京:科学出版社,1981:363-365
[30]Buehler R J.Confidence interval for the product of two Binomial parameters[J].J Amer Statist Assoc,1957,52:482-493
[31]陈家鼎.样本空间中的序与参数的置信限[J].数学进展,1993,22(6):542-552
[32]Wei zhongpeng,On the Confidence limits of the Reliability in the Case of I type Interval Censored Data[J],ACTA Mathematicae Applicatae Sinica,2006,29(1):81-90
[33]程侃著.寿命分布类与可靠性数学理论[M].北京:科学出版社,1999:494-509
[34]王瑞省.鞍点规划与模糊鞍点规划[D],河北:河北大学,2002
[35]Jens Ledet Jensen.Saddlepoint Approximations[M].New York:Oxford University Press,1995
[36]Dong Yan.The Lower Confidence limit of Reliability for Exponential life Data with Varying Censoring Time[J],ACTA Mathematicae Applicatae Sinica,2006,29(1):61-67
[37]Mao Shisong.Advanced Mathematical Statistics[M].Beijing China Higher Education Press;Berlin Heidelberg:Springer-Verlag,1998(in Chinese):271-295
[38]Lawless J F.Statistical Models and Methods for Lifetime Data[R].New York:John Wiley,1982
[39]Bartholomew D J.The Sampling Distribution of an Estimate Arising in Life Testing[J].Technometrics,1963,5(3):361-374
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