条件顺序统计量和样本间隔的随机比较以及应用
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摘要
在本论文中,我们对基于独立同分布和独立不同分布样本的条件顺序统计量以及样本间隔进行随机比较.这些研究结果可以应用于与概率统计相关的许多领域中,例如,寿命检验,生存分析,运筹学,经济,保险,精算,尤其是可靠性理论.
首先,我们集中研究n中取k系统的GRL(一般剩余寿命)和系统元件的GIT(一般休止时间),它们本质上是条件顺序统计量.研究基于GIT和GRL的随机比较问题,同时,我们也对它们的年龄性质进行调查.进一步,我们将研究工作往两个方向扩展.其一是在广义顺序统计量的框架下研究这些相应的问题,那么所得到的结果对于其它有序随机变量模型都是适用的,包括记录值模型,累进Ⅱ型截断顺序统计量等.其二,我们将研究有效的扩展到独立不同分布样本的情形,其中‘积和式'的使用是解决这个问题的关键所在.
其次,我们研究样本间隔的可靠性性质.众所周知,记录值的样本间隔实际上对应于非齐次泊松过程的到达点时间差(详见Belzunce 1999).那么,我们这里将研究基于单样本或者双样本的记录值间隔的随机比较问题.设X_1,…,X_p是具有失效率λ的指数样本,设X_(p+1),…,X_(p+q)是失效率为λ~*的又一指数样本,其中p+q=n.在这种指数框架下,我们建立了混合样本的一般间隔的似然比序结果.
再次,我们得到了非齐次指数样本的样本极差及第二顺序统计量的随机序的等价刻画.设X_1,…,X_n为独立的指数随机变量而且具有各自的失效率λ_1,…,λ_n,并且设Y_1,…,Y_n为一独立同分布的指数样本且具有共同的失效率λ.在这种设置下,样本极差X_(n,n)-X_(1,n)和Y_(n,n)-Y_(1,n)之间的通常随机序可以用其参数的不等式来等价刻画.另外,我们也建立了第二顺序统计量X_(2,n)和Y_(2,n)之间的似然比序的等价刻画.
最后,我们引进了基于比例机率的混合模型.在总体随机变量和不可观测的混合随机变量之间建立了TP2相依性;同时也给出了一些寿命分布类的封闭性结果.当混合变量或者是基本变量以某种随机序排列时,我们对其相应的总体变量进行了随机比较.另外,我们也找到了一个有用的概率界.
In this thesis,we will carry out stochastic comparison based on conditional order statistics and sample spacings from i.i.d,and non-i.i.d,samples,which has wide applications in various areas of applied probability and statistics including life testing,survival analysis,operations research,economics,insurance,actuarial sciences,reliability theory.
Firstly,we put our attention to GRL(general residual life) and GIT(general inactive time) of a(n-k+1)-out-of-n(or equivalently,conditional order statistics).We carry out stochastic comparison and examine aging properties based on GITs and GRLs.Further, the study is extended in two directions,one direction is based on the conditional generalized order statistics and hence some related results involved in other ordered random variables such as record values,progressively typeⅡcensored order statistics,are established as well;the other direction is based on the order statistics from independent but not identically distributed observations.
Secondly,we focus on the topic about spacings.It is well-known that spacings of record values inter-epoch intervals of nonhomogeneous Poisson processes,which can be regarded as spacings of record values(see Belzunce 1999).We will further carry out stochastic comparison of spacings of record values from one sample and two samples.Let X_1,…,X_p be a simple random sample from an exponential distribution with failure rateλ;Let X_(p+1),…,X_(p+q) be another simple random sample from the exponential distribution with failure rateλ~*,where p+q=n.Under this exponential framework,the likelihood ratio order between m-spacings of the combined sample is developed.
Thirdly,we establish characterization results in terms of parameters for stochastic orders of sample range and the second order statistics involved in heterogeneously exponential random variables.Under the heterogeneously exponential setup,we obtain the equivalent characterization in terms of parameters for the stochastic order between the sample range from nonidentical independent exponential observations and that from i.i.d,exponential ones;also,we derive the characterization for the likelihood ratio order between the second order statistics from heterogeneous exponential random variables.
Finally,we introduce a mixture model based on the proportional odds.We build the TP2 dependence between the population variable and the unobserved mixing variable and present some preservation properties.Relations on aging characteristics,such as odds function and(reversed) hazard rate are discussed.Stochastic comparisons on population variables are conducted as well.
首先,我们集中研究n中取k系统的GRL(一般剩余寿命)和系统元件的GIT(一般休止时间),它们本质上是条件顺序统计量.研究基于GIT和GRL的随机比较问题,同时,我们也对它们的年龄性质进行调查.进一步,我们将研究工作往两个方向扩展.其一是在广义顺序统计量的框架下研究这些相应的问题,那么所得到的结果对于其它有序随机变量模型都是适用的,包括记录值模型,累进Ⅱ型截断顺序统计量等.其二,我们将研究有效的扩展到独立不同分布样本的情形,其中‘积和式'的使用是解决这个问题的关键所在.
其次,我们研究样本间隔的可靠性性质.众所周知,记录值的样本间隔实际上对应于非齐次泊松过程的到达点时间差(详见Belzunce 1999).那么,我们这里将研究基于单样本或者双样本的记录值间隔的随机比较问题.设X_1,…,X_p是具有失效率λ的指数样本,设X_(p+1),…,X_(p+q)是失效率为λ~*的又一指数样本,其中p+q=n.在这种指数框架下,我们建立了混合样本的一般间隔的似然比序结果.
再次,我们得到了非齐次指数样本的样本极差及第二顺序统计量的随机序的等价刻画.设X_1,…,X_n为独立的指数随机变量而且具有各自的失效率λ_1,…,λ_n,并且设Y_1,…,Y_n为一独立同分布的指数样本且具有共同的失效率λ.在这种设置下,样本极差X_(n,n)-X_(1,n)和Y_(n,n)-Y_(1,n)之间的通常随机序可以用其参数的不等式来等价刻画.另外,我们也建立了第二顺序统计量X_(2,n)和Y_(2,n)之间的似然比序的等价刻画.
最后,我们引进了基于比例机率的混合模型.在总体随机变量和不可观测的混合随机变量之间建立了TP2相依性;同时也给出了一些寿命分布类的封闭性结果.当混合变量或者是基本变量以某种随机序排列时,我们对其相应的总体变量进行了随机比较.另外,我们也找到了一个有用的概率界.
In this thesis,we will carry out stochastic comparison based on conditional order statistics and sample spacings from i.i.d,and non-i.i.d,samples,which has wide applications in various areas of applied probability and statistics including life testing,survival analysis,operations research,economics,insurance,actuarial sciences,reliability theory.
Firstly,we put our attention to GRL(general residual life) and GIT(general inactive time) of a(n-k+1)-out-of-n(or equivalently,conditional order statistics).We carry out stochastic comparison and examine aging properties based on GITs and GRLs.Further, the study is extended in two directions,one direction is based on the conditional generalized order statistics and hence some related results involved in other ordered random variables such as record values,progressively typeⅡcensored order statistics,are established as well;the other direction is based on the order statistics from independent but not identically distributed observations.
Secondly,we focus on the topic about spacings.It is well-known that spacings of record values inter-epoch intervals of nonhomogeneous Poisson processes,which can be regarded as spacings of record values(see Belzunce 1999).We will further carry out stochastic comparison of spacings of record values from one sample and two samples.Let X_1,…,X_p be a simple random sample from an exponential distribution with failure rateλ;Let X_(p+1),…,X_(p+q) be another simple random sample from the exponential distribution with failure rateλ~*,where p+q=n.Under this exponential framework,the likelihood ratio order between m-spacings of the combined sample is developed.
Thirdly,we establish characterization results in terms of parameters for stochastic orders of sample range and the second order statistics involved in heterogeneously exponential random variables.Under the heterogeneously exponential setup,we obtain the equivalent characterization in terms of parameters for the stochastic order between the sample range from nonidentical independent exponential observations and that from i.i.d,exponential ones;also,we derive the characterization for the likelihood ratio order between the second order statistics from heterogeneous exponential random variables.
Finally,we introduce a mixture model based on the proportional odds.We build the TP2 dependence between the population variable and the unobserved mixing variable and present some preservation properties.Relations on aging characteristics,such as odds function and(reversed) hazard rate are discussed.Stochastic comparisons on population variables are conducted as well.
引文
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