基于多元copula函数的串联系统应力-强度模型可靠度的非参数估计

基于多元copula函数的串联系统应力-强度模型可靠度的非参数估计 Nonparametric Estimation for the Reliability of a Series System Stress-Strength Model Based on Multiple Copula Function 1 first-author 祁辉，男，副教授，主要从事生存分析与可靠性方面的研究. E-mail：qh19810@126.com 祁辉，男，副教授，主要从事生存分析与可靠性方面的研究. E-mail： qh19810@126.com qh19810@126.com 2 三明学院 信息工程学院/数字福建工业能源大数据研究所，福建 三明 365004 三明学院 信息工程学院/数字福建工业能源大数据研究所，福建 三明 365004 Institute of Information Engineering/ Digital Fujian Research Institute for Industrial Energy Big Data, Sanming University, Sanming 365004, Fujian, China Institute of Information Engineering/ Digital Fujian Research Institute for Industrial Energy Big Data, Sanming University, Sanming 365004, Fujian, China 武汉大学 数学与统计学院，湖北 武汉 430072 武汉大学 数学与统计学院，湖北 武汉 430072 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China School of Mathematics and Statistics, Wuhan University, Wuhan 430072, Hubei, China 针对多部件的串联系统应力-强度模型可靠度的估计问题, 利用多元Farlie-Gumbel-Morgenstern copula(FGM)函数或多元Clayton copula函数来度量变量之间的相关性, 给出了在变量非删失或右删失情形下模型可靠度的非参数估计, 并证明了估计量的渐近性质.数值模拟的结果表明该方法在有限样本下表现良好. To estimate the reliability of a series system the stress-strength model with multiple components, we utilize the multivariate Farlie-Gumbel-Morgenstern(FGM) copula function or multivariate Clayton copula function to measure the correlations of variables. We obtained the nonparametric estimation for the reliability under no censored or right censored cases and verified the asymptotic properties of the resultant estimators. Simulation results show that the proposed method performs well in finite samples. 应力-强度模型 连接函数 Kaplan-Meier估计量 右删失 stress-strength model copula function Kaplan-Meier estimator right censored 引言 1 在可靠性系统中, 应力-强度模型假定随机变量 Y 为系统部件的强度, 随机变量 X 表示外界施加于系统的应力, 当随机应力大于系统部件的强度时, 系统便失效.因此应力-强度模型可靠度 R = P ( X < Y )可以很好地度量系统的性能.关于模型可靠度的统计推断是众多研究者普遍关注的问题.该模型在航空航天、机械工程、土木工程和临床试验等领域有着广泛的应用. 1956年Birnbaum [ 1 1 ] 在Berkeley概率统计学术研讨会上提出了关于应力-强度模型可靠度的估计问题.此后, 该问题便引起众多研究者的兴趣.研究成果多集中于讨论当随机应力变量 X 和强度变量 Y 服从各种分布时模型可靠度的估计, 详见文献[ 2 2 ]. Abid和Hassan [ 3 3 ] 讨论了当应力变量和强度变量相对独立并均服从MOEU分布时模型可靠度的极大似然估计和加权最小二乘估计等估计方法.基于极大似然估计和Bootstrap方法, Ghitany等 [ 4 4 ] 研究了应力变量和强度变量相互独立并均服从Power Lindley分布时 R 的点估计和区间估计. Sharma等 [ 5 5 ] 对相互独立的应力变量和强度变量均服从Inverse Lindley分布时模型可靠度 R 进行了估计.类似地, Baklizi [ 6 6 ] 研究了两参数均服从指数分布时 R 的区间估计. 以上研究主要是对由单个部件所构成的系统应力-强度模型可靠度的估计.对于由 n 个部件所构成的串联结构系统, Chandra和Owen [ 7 7 ] 分别讨论了在正态和指数分布情形下, 部件的随机强度变量分别为 Y 1 , Y 2 , …, Y k , 随机应力变量为 X 时系统应力-强度模型可靠度 P ( X < Y 1 , X < Y 2 , …, X < Y k )的极大似然估计和一致最小无偏估计.考虑到应力变量相关的情形, Hanagal [ 8 8 ] 对强度变量服从二元指数分布, 随机应力变量服从指数或Gamma分布的二元并联系统应力-强度模型可靠度进行了估计.类似的工作有Hanagal [ 9 9 ] 对于二元串联系统应力-强度模型可靠度的估计.叶慈南 [ 10 10 ] 研究了强度变量服从二元Mobve分布, 应力变量服从指数分布的二元并联系统应力-强度模型可靠度的两种估计量及其渐近置信下限.针对应力变量和强度变量相互独立并均服从Gamma, Weibull以及Pareto分布的串联结构系统, Hanagal [ 11 11 ] 给出了在上述情形下系统应力-强度模型可靠度的估计.王炳兴 [ 12 12 ] 提出了应力变量和强度变量相互独立并均服从指数分布时, 串联结构系统应力-强度模型可靠度的广义区间估计.基于模糊集理论, Eryilmaz和Tütüncü [ 13 13 ] 研究了串联结构系统应力-强度模型可靠度的估计. 现实中随机应力变量和强度变量之间存在相关性.一个简单的例子，如比较两个系统a和b的有效性, 系统a由应力变量为 X 的单个部件构成, 系统b由强度变量为 Y 和 Z 的两个部件串联构成.假定三者具有相同的动力装置, 因此对模型可靠度 P ( X < min( Y , Z ))的估计需考虑变量之间的相关性. copula函数类广泛应用于变量之间相关性的度量, 其表示变量之间的相关关系, 不仅局限于线性相关, 还可以表示变量之间的非线性相关. Domma [ 14 14 ] 将copula函数用于应力变量和强度变量之间相关性的度量, 从而实现当变量服从Burr等分布时模型可靠度 P ( X < Y )的估计.有鉴于此, 本文提出了基于多元copula函数的串联结构系统应力-强度模型可靠度的非参数估计, 其中变量之间的相关性采用常见的多元Farlie-Gumbel-Morgenstern (FGM) copula函数或多元Clayton copula函数来度量.这种非参数的估计方法避免了变量分布未知时给模型可靠度的估计带来的困难.进一步, 考虑到在可靠性数据收集过程中, 由于观测对象进入或退出观察时间的差别, 从而导致样本出现删失的情形，Qi等 [ 15 15 ] 考虑了相互独立的变量在右删失情形下 P ( X < Y )的估计.在此基础上, 本文也研究了样本右删失情形下上述模型可靠度的估计. 基于多元FGM copula函数的串联系统可靠度的估计 1 任何具有边缘分布函数 F 1 , F 2 , …, F k 的 k 维随机向量, 一定存在一个 k 元copula函数 C , 使得该向量的分布函数可表示成: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258250&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258250&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258250&type=middle 其中, 向量的联合概率密度函数为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258263&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258263&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258263&type=middle 近一步, 当边缘分布函数 F 1 , F 2 , …, F k 均连续时, 那么copula函数 C 惟一.copula函数能充分描述变量之间的相关性结构.因此, 利用copula函数可以使得构造相关的多元随机变量的联合分布变得更加容易.关于copula函数理论的详细介绍, 可以参考文献[ 16 16 ]. FGM copula函数由于简单的结构和优良的分析性质而在统计建模中有着广泛的应用, 其 k 元函数的表达式为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258276&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258276&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258276&type=middle 3元FGM copula函数为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258288&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258288&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258288&type=middle 对于2元的串联结构系统, 若系统部件的强度变量分别为 Y 和 Z , 所受外界的随机应力变量为 X , 并假定以上变量均为非负的随机变量, 如前所述, 随机应力变量 X 和部件强度变量 Y 和 Z 之间可能存在相关性.如果我们用3元FGM copula函数来度量这种相关性, 考虑到将问题简化, 取3元FGM copula函数的参数 θ 12 、 θ 13 以及 θ 23 均为零, 则系统应力-强度模型可靠度为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258297&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258297&type=middle 其中函数 F X ( t ), F Y ( t ), F Z ( t )分别为变量 X , Y , Z 的分布函数.在一个容量为 n 的样本( X i , Y i , Z i ) n i =1 下, 可分别用: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258305&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258305&type=middle 来估计 F X ( t ), F Y ( t ), F Z ( t )，从而可得 R 12 的估计如下: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258312&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258312&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258312&type=middle 考虑到变量删失的情形, 首先定义变量 X , Y , Z 的右删失观测变量以及删失指标函数如下: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258321&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258321&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258321&type=middle 其中, 随机变量 C , L , M 分别为 X , Y , Z 的右删失变量, 则样本可表示为( T i , Δ i , K i , Q i , D i , S i ) i =1 n .由样本提供的信息, 我们构造变量 X , Y , Z 的分布函数的Kaplan-Meier估计量如下: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258328&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258328&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258328&type=middle 将上述估计量代入(1)式, 我们可得在变量右删失情形下模型可靠度的估计: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258334&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258334&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258334&type=middle 利用 k +1元FGM copula函数来度量变量之间的相关关系, 上述结论可以推广到 k 元相依的串联结构系统应力-强度模型可靠度的估计. 基于多元Clayton copula函数的串联结构系统可靠度的估计 1 Clayton copula函数是一种重要的阿基米德copula函数, 其在统计建模中经常用于随机变量之间相关性的度量. k 元Clayton copula函数的一般表达式为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258343&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258343&type=middle 其中 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258432&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258432&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258432&type=middle 为其母函数. 3元Clayton copula函数表达式为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258349&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258349&type=middle 类似的, 对于2元串联结构系统, 如果用3元Clayton copula函数来度量应力变量 X 以及强度变量 Y 和 Z 之间的相关性，则系统应力-强度模型可靠度可表示为： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258355&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258355&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258355&type=middle 同上所述, 将变量 X , Y , Z 的分布函数的估计量代入(4)式, 从而可得2元串联结构系统应力-强度模型可靠度的非参数估计: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258362&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258362&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258362&type=middle 考虑到变量右删失的情形, 则基于3元Clayton copula函数的应力-强度模型可靠度为: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258369&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258369&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258369&type=middle 类似的, 若利用 k +1元Clayton copula函数来度量变量之间的相关关系, 上述结论也可以推广到 k 元相依的串联结构系统应力-强度模型可靠度的估计. 估计量的渐近性质 1 下面证明上述可靠度估计量的渐近性质.首先我们给出以下假定条件: C1  假定应力变量 X 是有界的, 其取值区间为[0, τ ], 分布函数 F X ( t )关于变量 t 是连续的. C2  存在常数 c 0 > 0, 使得 P ( C ≥ τ )≥ c 0 , 且函数 P ( C > t )关于变量 t 连续. C3  假定强度变量 Y 也是有界的, 其取值区间为[0, τ ], 分布函数 F Y ( t )关于变量 t 是连续的. C4  存在常数 l 0 > 0, 使得 P ( L ≥ τ )≥ l 0 , 且函数 P ( L > t )关于变量 t 连续. C5  假定强度变量 Z 也有上述相同的限定, 且分布函数 F Z ( t )关于变量 t 是连续的. C6  存在常数 m 0 > 0, 使得 P ( M ≥ τ )≥ m 0 , 且函数 P ( M > t )关于变量 t 连续. 在右删失情形下, 基于FGM copula函数的2元串联系统应力-强度模型可靠度的估计量可表示成如下的形式： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258376&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258376&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258376&type=middle 其中，3元函数 g 3 (·)是光滑的连续函数.在上述条件下, 可以得到如下结论: 引理1 [ 17 17 ]  由条件C1和C2可得, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258455&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258455&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258455&type=middle .由条件C3、C4和C5、C6可知, 类似的结论对估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type=middle 也成立. 由引理1和连续映射定理, 可知： http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258381&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258381&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258381&type=middle 引理2 [ 18 18 ]  由条件C1和C2可得, 在区间[0, τ ]上, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258489&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258489&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258489&type=middle .其中 W X ( t )为零均值的连续路径高斯过程.类似的, 该结论对 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258495&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258504&type=middle 也成立. 由引理2和Delta方法 [ 19 19 ] , 在区间[0, τ ]上, 可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258386&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258386&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258386&type=middle 其中, W 3 ( t )为零均值的连续路径高斯过程. 定理1  由条件C1和C6, 当样本右删失时, 基于FGM copula函数的2元串联系统应力-强度模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 在样本容量 n →+∞时, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258519&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258519&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258519&type=middle .当样本非删失时, 模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 也满足上述渐近性质. 证  将 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258551&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258551&type=middle 写成如下的一般形式: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258388&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258388&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258388&type=middle 由引理1和引理2, 可得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258390&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258390&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258390&type=middle 利用强嵌入定理 [ 20 20 ] , 可得存在一个新的概率空间, 使得: http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258392&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258392&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258392&type=middle 由文献[ 21 21 ], 可得, 对任意 t ∈ [0, τ ] http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258395&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258395&type=middle 因此, 在新的概率空间下, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258403&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258403&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258403&type=middle 所以, 在原概率空间下, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258560&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258560&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258560&type=middle 由于非删失情形下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 是估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 在删失率为零时的特殊情况, 因此以上渐近性质对估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 同样适用.定理得证. 定理2  由条件C1和C6, 当样本右删失时, 基于Clayton copula函数的2元串联结构系统应力-强度模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258588&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258588&type=middle 在样本容量 n →+∞时, http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258597&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258597&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258597&type=middle . 当样本非删失时, 模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 也满足上述渐近性质.定理的证明同上, 故略. 类似的, 以上渐近性质可以推广到 k 元串联结构系统应力-强度模型可靠度的估计量. 随机模拟 1 下面主要研究当利用3元FGM copula函数或3元Clayton copula函数来度量变量之间的相关性时, 在变量非删失或右删失的情形下, 上述关于2元串联结构系统应力-强度模型可靠度的估计量在有限样本下的统计表现. 基于FGM copula函数的应力-强度模型可靠度的估计量的随机模拟 2 对于2元串联结构系统应力-强度模型, 当变量之间的相关性采用3元FGM copula函数来度量时, 首先假定应力变量 X 以及强度变量 Y 和 Z 的边缘分布分别服从参数为 λ 1 , λ 2 , λ 3 的指数分布, 其中( λ 1 , λ 2 , λ 3 )取值包括如下四种情形:(2, 2, 2), (1, 2, 3), (3, 1, 2), (2, 3, 1).分别从以( λ 1 , λ 2 , λ 3 )为参数的指数联合总体中抽取样本量 n 为50到200之间的大小不等的样本, 从而计算出模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 在不同样本下的统计指标, 如:估计量的估计值(Est), 估计量的标准差(SD), 标准差的估计值(SE), 95%置信区间覆盖率(CP), 其中SE由bootstrap方法 [ 22 22 ] 从原样本中重复抽样 n 次计算得到, 并通过SE构造相应的置信区间.同时, 考虑到变量 X , Y , Z 右删失的情形, 删失变量 C , L , M 均由指数分布exp(1)生成.此外, FGM copula函数的参数 θ 取值体现了变量之间相关性的差异. θ 取较小的值-0.8, 基于500次数值模拟的结果见 表 1 表 1 ； θ 取较大的值0.8, 基于500次数值模拟的结果见 表 2 表 2 . Censoring rate n λ 1 λ 2 λ 3 R Est SD SE CP 3 (0.000, 0.000, 0.000) 50 2.000 2.000 2.000 0.333 0.338 0.053 0.052 0.934 100 2.000 2.000 2.000 0.333 0.333 0.038 0.037 0.938 200 2.000 2.000 2.000 0.333 0.333 0.027 0.027 0.948 3 (0.328, 0.337, 0.329) 50 2.000 2.000 2.000 0.333 0.338 0.059 0.055 0.912 100 2.000 2.000 2.000 0.333 0.332 0.040 0.040 0.936 200 2.000 2.000 2.000 0.333 0.333 0.028 0.028 0.962 3 (0.000, 0.000, 0.000) 50 1.000 2.000 3.000 0.160 0.162 0.040 0.040 0.924 100 1.000 2.000 3.000 0.160 0.160 0.029 0.028 0.930 200 1.000 2.000 3.000 0.160 0.159 0.021 0.020 0.924 3 (0.506, 0.328, 0.249) 50 1.000 2.000 3.000 0.160 0.160 0.044 0.042 0.914 100 1.000 2.000 3.000 0.160 0.158 0.029 0.030 0.936 200 1.000 2.000 3.000 0.160 0.159 0.023 0.021 0.932 3 (0.000, 0.000, 0.000) 50 3.000 1.000 2.000 0.507 0.503 0.057 0.056 0.920 100 3.000 1.000 2.000 0.507 0.506 0.039 0.040 0.946 200 3.000 1.000 2.000 0.507 0.506 0.028 0.028 0.956 3 (0.246, 0.497, 0.332) 50 3.000 1.000 2.000 0.507 0.506 0.062 0.059 0.926 100 3.000 1.000 2.000 0.507 0.503 0.042 0.043 0.930 200 3.000 1.000 2.000 0.507 0.506 0.031 0.030 0.950 3 (0.000, 0.000, 0.000) 50 2.000 3.000 1.000 0.333 0.332 0.057 0.052 0.910 100 2.000 3.000 1.000 0.333 0.334 0.037 0.038 0.958 200 2.000 3.000 1.000 0.333 0.335 0.028 0.027 0.924 3 (0.327, 0.242, 0.495) 50 2.000 3.000 1.000 0.333 0.337 0.058 0.056 0.926 100 2.000 3.000 1.000 0.333 0.334 0.040 0.040 0.938 200 2.000 3.000 1.000 0.333 0.334 0.030 0.028 0.946 Censoring rate n λ 1 λ 2 λ 3 R Est SD SE CP 50 2.000 2.000 2.000 0.333 0.335 0.052 0.049 0.916 (0.000, 0.000, 0.000) 100 2.000 2.000 2.000 0.333 0.329 0.035 0.035 0.938 200 2.000 2.000 2.000 0.333 0.333 0.024 0.025 0.950 50 2.000 2.000 2.000 0.333 0.335 0.055 0.052 0.910 (0.327, 0.338, 0.328) 100 2.000 2.000 2.000 0.333 0.331 0.036 0.037 0.948 200 2.000 2.000 2.000 0.333 0.332 0.026 0.026 0.932 50 1.000 2.000 3.000 0.174 0.173 0.039 0.038 0.910 (0.000, 0.000, 0.000) 100 1.000 2.000 3.000 0.174 0.175 0.028 0.027 0.930 200 1.000 2.000 3.000 0.174 0.174 0.019 0.019 0.944 50 1.000 2.000 3.000 0.174 0.171 0.040 0.040 0.930 (0.493, 0.329, 0.241) 100 1.000 2.000 3.000 0.174 0.174 0.028 0.029 0.944 200 1.000 2.000 3.000 0.174 0.175 0.020 0.020 0.946 50 3.000 1.000 2.000 0.493 0.495 0.053 0.054 0.942 (0.000, 0.000, 0.000) 100 3.000 1.000 2.000 0.493 0.495 0.040 0.039 0.932 200 3.000 1.000 2.000 0.493 0.495 0.028 0.027 0.952 50 3.000 1.000 2.000 0.493 0.506 0.062 0.059 0.936 (0.252, 0.504, 0.334) 100 3.000 1.000 2.000 0.493 0.503 0.042 0.043 0.952 200 3.000 1.000 2.000 0.493 0.506 0.031 0.030 0.954 50 2.000 3.000 1.000 0.333 0.334 0.053 0.050 0.920 (0.000, 0.000, 0.000) 100 2.000 3.000 1.000 0.333 0.334 0.037 0.036 0.938 200 2.000 3.000 1.000 0.333 0.331 0.028 0.025 0.948 50 2.000 3.000 1.000 0.333 0.330 0.053 0.053 0.930 (0.340, 0.264, 0.509) 100 2.000 3.000 1.000 0.333 0.334 0.039 0.048 0.946 200 2.000 3.000 1.000 0.333 0.331 0.027 0.027 0.930 通过 表 1 表 1 , 可以发现, 当样本量较小取 n =50时, 考虑到参数 λ 1 , λ 2 以及 λ 3 取值的四种不同情形, 样本非删失情形下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 以及样本右删失情形下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 的各项统计指标均表现良好.估计值与其真值偏差较小, 标准差的估计值接近其真值, 由bootstrap方法构造的95%置信区间覆盖率在95%附近.此外, 通过对比参数 λ 1 , λ 2 以及 λ 3 取值的每一种情形下估计量的统计指标, 我们可以发现, 在样本量 n 相同, 样本右删失下的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 与非删失下的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 的各项统计指标比较接近.当样本量 n 从50增加到200时, 删失与非删失情形下的估计值更接近真值, 标准差的估计变小, 95%置信区间覆盖率一般会更接近95%. 此外, 表 2 表 2 的随机模拟结果显示:当变量之间的相关性在另一种情形下, 即 θ 的取值变化为0.8时，估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258686&type=middle 与 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258694&type=middle 依然表现良好. 基于Clayton copula函数的应力-强度模型可靠度的估计量的随机模拟 2 当变量之间的相关性采用Clayton copula函数来度量时, 对于样本非删失情形下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 以及样本右删失情形下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 的随机模拟研究中, 随机模拟条件:变量边缘分布的假定, 参数的选择, 样本量大小的选定, 删失变量的处理, 重复计算的次数等, 都与上述模拟实验假定相同.考虑到Clayton copula函数的参数 θ 取值大小体现了变量之间相关性的强弱，因此, 我们分别考虑了 θ =0.8, 即变量之间相关性较弱时, 估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 的随机模拟，结果见 表 3 表 3 ； θ =2时, 即变量之间的相关性较强时, 估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 的随机模拟，结果见 表 4 表 4 . Censoring rate n λ 1 λ 2 λ 3 R Est SD SE CP 50 2.000 2.000 2.000 0.334 0.342 0.053 0.053 0.960 (0.000, 0.000, 0.000) 100 2.000 2.000 2.000 0.334 0.339 0.039 0.039 0.936 200 2.000 2.000 2.000 0.334 0.336 0.026 0.028 0.964 50 2.000 2.000 2.000 0.334 0.348 0.064 0.059 0.916 (0.342, 0.336, 0.335) 100 2.000 2.000 2.000 0.334 0.342 0.046 0.042 0.920 200 2.000 2.000 2.000 0.334 0.336 0.030 0.030 0.940 50 1.000 2.000 3.000 0.096 0.113 0.032 0.034 0.946 (0.000, 0.000, 0.000) 100 1.000 2.000 3.000 0.096 0.107 0.024 0.024 0.942 200 1.000 2.000 3.000 0.096 0.104 0.017 0.017 0.930 50 1.000 2.000 3.000 0.096 0.120 0.037 0.037 0.932 (0.509, 0.329, 0.252) 100 1.000 2.000 3.000 0.096 0.107 0.027 0.026 0.914 200 1.000 2.000 3.000 0.096 0.103 0.017 0.018 0.954 50 3.000 1.000 2.000 0.574 0.585 0.060 0.056 0.916 (0.000, 0.000, 0.000) 100 3.000 1.000 2.000 0.574 0.585 0.043 0.040 0.922 200 3.000 1.000 2.000 0.574 0.582 0.029 0.029 0.946 50 3.000 1.000 2.000 0.574 0.584 0.065 0.061 0.920 (0.260, 0.499, 0.343) 100 3.000 1.000 2.000 0.574 0.584 0.047 0.044 0.930 200 3.000 1.000 2.000 0.574 0.580 0.032 0.032 0.940 50 2.000 3.000 1.000 0.324 0.334 0.054 0.053 0.930 (0.000, 0.000, 0.000) 100 2.000 3.000 1.000 0.324 0.328 0.039 0.038 0.950 200 2.000 3.000 1.000 0.324 0.328 0.029 0.027 0.924 50 2.000 3.000 1.000 0.324 0.338 0.063 0.059 0.924 (0.323, 0.245, 0.500) 100 2.000 3.000 1.000 0.324 0.331 0.043 0.042 0.950 200 2.000 3.000 1.000 0.324 0.325 0.029 0.030 0.960 Censoring rate n λ 1 λ 2 λ 3 R Est SD SE CP 50 2.000 2.000 2.000 0.331 0.360 0.055 0.056 0.910 (0.000, 0.000, 0.000) 100 2.000 2.000 2.000 0.331 0.346 0.042 0.040 0.930 200 2.000 2.000 2.000 0.331 0.341 0.030 0.029 0.932 50 2.000 2.000 2.000 0.331 0.359 0.066 0.062 0.916 (0.336, 0.338, 0.342) 100 2.000 2.000 2.000 0.331 0.351 0.046 0.046 0.912 200 2.000 2.000 2.000 0.331 0.342 0.032 0.033 0.944 50 1.000 2.000 3.000 0.036 0.055 0.021 0.024 0.954 (0.000, 0.000, 0.000) 100 1.000 2.000 3.000 0.036 0.046 0.014 0.015 0.966 200 1.000 2.000 3.000 0.036 0.042 0.010 0.010 0.946 50 1.000 2.000 3.000 0.036 0.057 0.025 0.029 0.966 (0.494, 0.328, 0.242) 100 1.000 2.000 3.000 0.036 0.048 0.016 0.018 0.962 200 1.000 2.000 3.000 0.036 0.042 0.011 0.012 0.958 50 3.000 1.000 2.000 0.674 0.678 0.054 0.054 0.928 (0.000, 0.000, 0.000) 100 3.000 1.000 2.000 0.674 0.676 0.041 0.039 0.924 200 3.000 1.000 2.000 0.674 0.677 0.029 0.028 0.930 50 3.000 1.000 2.000 0.674 0.684 0.065 0.061 0.912 (0.248, 0.493, 0.318) 100 3.000 1.000 2.000 0.674 0.680 0.043 0.044 0.948 200 3.000 1.000 2.000 0.674 0.675 0.032 0.032 0.942 50 2.000 3.000 1.000 0.291 0.316 0.057 0.054 0.910 (0.000, 0.000, 0.000) 100 2.000 3.000 1.000 0.291 0.304 0.036 0.039 0.956 200 2.000 3.000 1.000 0.291 0.298 0.028 0.027 0.954 50 2.000 3.000 1.000 0.291 0.320 0.060 0.062 0.928 (0.334, 0.252, 0.505) 100 2.000 3.000 1.000 0.291 0.313 0.043 0.043 0.918 200 2.000 3.000 1.000 0.291 0.301 0.032 0.031 0.922 通过 表 3 表 3 , 可以发现, 当采用Clayton copula函数来度量变量之间的相关性时, 样本非删失下模型可靠度的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 与右删失下的估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 仍具有良好的性质，即当样本量较小时, 两者估计值与真值偏差较小, 标准差的估计值均接近真值, 由bootstrap方法估计的95%置信区间覆盖率在95%附近.对于参数 λ 1 , λ 2 以及 λ 3 取值的每一种情形, 可以看出, 随着样本量 n 的增大, 估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 的各项统计指标均表现得更好. 通过 表 4 表 4 , 也可以得出, 即使改变变量之间的相关性, 让变量之间相关性变强时, 估计量 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258815&type=middle 和 http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=small http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4258818&type=middle 依然有良好的表现. 综上所述, 通过考察我们提出的估计量在各种不同情形下的随机模拟结果, 验证了这些估计量具有良好的统计性质. 结论 1 本文研究了相关的应力变量和强度变量在非删失及右删失的情形下, 二元串联结构系统应力-强度模型可靠度的非参数估计.其中，变量之间的相关性采用3元FGM copula函数或3元Clayton copula函数来度量.上述方法也可以推广到对 k 元相依的串联结构系统应力-强度模型可靠度的估计.采用经验过程的理论, 本文对所提出的估计量的渐近性质进行了证明.通过考察估计量在不同情形下的随机模拟结果, 验证了估计量具有良好的统计性质.此外, 上述结论也可以推广到 k 元相依的并联结构系统应力-强度模型可靠度的估计. 参考文献 BIRNBAUM Z W. 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Empirical Processes with Applications to Statistics [M]. New York: Wiley, 1986. LIN D Y, WEI L J, YANG I, et al . Semiparametric regression for the mean and rate functions of recurrent events [J]. Journal of the Royal Statistical Society : Series B , 2000, 62 (4):711-730.DOI:10.1111/1467-9868.00259. RICE J A. Mathematical Statistics and Data Analysis [M]. 3rd ed. Massachusetts: Duxbury Resource Center, 2006. 10.14188/j.1671-8836.2018.03.011 O213.2 whdxxblxb-64-3-269 国家自然科学基金(11401341)；福建省自然科学基金(2016J01681)；福建省教育厅中青年基金(JAT160468)；福建省高校杰出青年科研人才培育计划(2015)，福建省高校新世纪优秀人才支持计划([2016]23号)资助项目 2017-10-09 2018-06-24 2020-11-30 版权所有© 2018 《武汉大学学报(理学版)》编辑部 Copyright © 2018 Journal of Wuhan University(Natural Science Edition). All rights reserved. 2018 武汉大学学报(理学版) 3 64