异质模型与顺序统计量中的随机比较
摘要
在本论文中,我们研究了异质模型(Frailty models)和基于独立非齐次指数样本的顺序统计量中的随机比较问题.
异质模型作为处理不可观测个体差异的有效工具,在过去的二三十年中,已经被广泛的应用于流行病学、人口统计学以及与生存分析相关的其它领域中.在本论文的第一部分,我们分别考虑了几类异质模型中的随机比较.
首先,在经典的一元脆性异质模型(Vaupel et al.,1979)中,我们对总体寿命和具有特定异质因子的个体寿命进行了随机比较,建立了这两个寿命之间的一些常见随机序关系基于异质因子和异质变量矩的等价刻画,基于这些结果,我们讨论了风险评估中高风险组与总体之间的寿命比较.
其次,在广义脆性异质模型(Gupta&Gupta,2009)中,我们通过随机比较探讨了异质变量的不同选取(变化)对总体造成的影响.特别地,我们指出并纠正了Gupta& Gupta(2009)中一个错误结果.
再次,我们考虑了服从多元脆性异质模型的随机变量组之间的生存竞争问题,给出了描述竞争系统性能的评价指标的计算公式,通过随机比较考查了随机变量组的异质变量的变化对竞争系统的一些主要指标的影响.
最后,从一元比例反失效率异质模型出发,分别考虑了一般的基于反失效率建立的异质模型以及多元比例反失效率异质模型.在前者中,我们讨论了总体(混合)反失效率的单调性质以及总体休止时间,并建立了年龄性质DRHR在混合下封闭的充分条件；对于后者,我们讨论了异质向量的一些常见的随机序关系以及基底向量的年龄性质在混合下的封闭性,并进一步考虑了总体向量关于异质向量的随机单调性质以及总体多元反失效率的一些性质.
非齐次样本的顺序统计量在可靠性、生存分析、运筹学、精算等学科中具有重要的意义.在论文的第二部分,我们考虑了基于非齐次独立指数样本的顺序统计量的随机比较.首先,建立了基于非齐次指数样本和齐次指数样本的第二顺序统计量(故障安全系统)之间的右扩展序的等价刻画；其次,得到了两非齐次指数元件构成的并联系统的失效率序的充分条件,并进一步将该结果推广到比例失效率的情形.特别地,我们的结果很好的回答了Joo&Mi(2010)提出的一个公开问题.另外,对于元件具有一般分布的情形也做了一些讨论.
In this thesis, we have a thorough study on stochastic comparisons in frailty models as well as order statistics from independent heterogeneous exponential random variables.
As a particularly useful tool for handing heterogeneity left unexplained by observed covariates, during recent decades, frailty models have been widely applied in such as epidemiology, demography and other areas related to survival analysis. In the first part of this thesis, we focus on stochastic comparisons in several frailty models.
Firstly, in classical frailty models introduced by Vaupel et al. (1979), we conduct stochastic comparisons of the overall population and an individual with specific frailty, and establish equivalent characterizations for some well known stochastic orders between them in terms of the frailty and moments of the frailty variable. Upon these results we further compare the high risk groups with the overall population in risk assessment.
Secondly, in the general frailty model due to Gupta and Gupta (2009), the overall populations arising from different choices of the distribution of the frailty variable are compared. In particular, we correct a false result of Gupta and Gupta (2009).
Thirdly, we study the competing system between two groups of observations, each of them follows multivariate frailty models. Calculation formulas for some evaluation indices of the system are derived, and how the variation of the frailty vector has an impact on the performance of this system is investigated as well.
Lastly, based on univariate proportional reversed hazard rate frailty model, we have a further study on two kinds of frailty models, one is the univariate general frailty model in terms of the reversed hazard rate, the other is the multivariate proportional reversed hazard rate frailty model. For the former, some monotone properties of the overall pop-ulation (mixture) reversed hazard rate and the population inactivity time are discussed. Besides, a sufficient condition for the preservation of DRHR under the mixture is derived as well. For the latter, we investigate how some of the well known stochastic orders of the frailty vector and some ageing properties of the baseline are translated into those of the overall population. Also, the stochastic monotone properties of the overall population vector with respect to the frailty vector are discussed. And the overall population reversed hazard rate is discussed as well.
Order statistics from heterogeneous observations play an important role in various areas related to applied probability and statistics such as reliability theory, survival anal-ysis, operations research and actuarial sciences. In the second part, we conduct stochastic comparisons on order statistics from independent heterogeneous exponential random vari-ables.
Firstly, we establish equivalent characterizations in terms of parameters for the right spread ordering between the second order statistic (Fail-Safe system) from nonidentical independent exponential observations and that from i.i.d. exponential ones.
Secondly, we build a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent but nonidentical exponential components, this also serves as an answer to the open problem of Joo and Mi (2010). Moreover, this result is extended to the case with proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.
异质模型作为处理不可观测个体差异的有效工具,在过去的二三十年中,已经被广泛的应用于流行病学、人口统计学以及与生存分析相关的其它领域中.在本论文的第一部分,我们分别考虑了几类异质模型中的随机比较.
首先,在经典的一元脆性异质模型(Vaupel et al.,1979)中,我们对总体寿命和具有特定异质因子的个体寿命进行了随机比较,建立了这两个寿命之间的一些常见随机序关系基于异质因子和异质变量矩的等价刻画,基于这些结果,我们讨论了风险评估中高风险组与总体之间的寿命比较.
其次,在广义脆性异质模型(Gupta&Gupta,2009)中,我们通过随机比较探讨了异质变量的不同选取(变化)对总体造成的影响.特别地,我们指出并纠正了Gupta& Gupta(2009)中一个错误结果.
再次,我们考虑了服从多元脆性异质模型的随机变量组之间的生存竞争问题,给出了描述竞争系统性能的评价指标的计算公式,通过随机比较考查了随机变量组的异质变量的变化对竞争系统的一些主要指标的影响.
最后,从一元比例反失效率异质模型出发,分别考虑了一般的基于反失效率建立的异质模型以及多元比例反失效率异质模型.在前者中,我们讨论了总体(混合)反失效率的单调性质以及总体休止时间,并建立了年龄性质DRHR在混合下封闭的充分条件；对于后者,我们讨论了异质向量的一些常见的随机序关系以及基底向量的年龄性质在混合下的封闭性,并进一步考虑了总体向量关于异质向量的随机单调性质以及总体多元反失效率的一些性质.
非齐次样本的顺序统计量在可靠性、生存分析、运筹学、精算等学科中具有重要的意义.在论文的第二部分,我们考虑了基于非齐次独立指数样本的顺序统计量的随机比较.首先,建立了基于非齐次指数样本和齐次指数样本的第二顺序统计量(故障安全系统)之间的右扩展序的等价刻画；其次,得到了两非齐次指数元件构成的并联系统的失效率序的充分条件,并进一步将该结果推广到比例失效率的情形.特别地,我们的结果很好的回答了Joo&Mi(2010)提出的一个公开问题.另外,对于元件具有一般分布的情形也做了一些讨论.
In this thesis, we have a thorough study on stochastic comparisons in frailty models as well as order statistics from independent heterogeneous exponential random variables.
As a particularly useful tool for handing heterogeneity left unexplained by observed covariates, during recent decades, frailty models have been widely applied in such as epidemiology, demography and other areas related to survival analysis. In the first part of this thesis, we focus on stochastic comparisons in several frailty models.
Firstly, in classical frailty models introduced by Vaupel et al. (1979), we conduct stochastic comparisons of the overall population and an individual with specific frailty, and establish equivalent characterizations for some well known stochastic orders between them in terms of the frailty and moments of the frailty variable. Upon these results we further compare the high risk groups with the overall population in risk assessment.
Secondly, in the general frailty model due to Gupta and Gupta (2009), the overall populations arising from different choices of the distribution of the frailty variable are compared. In particular, we correct a false result of Gupta and Gupta (2009).
Thirdly, we study the competing system between two groups of observations, each of them follows multivariate frailty models. Calculation formulas for some evaluation indices of the system are derived, and how the variation of the frailty vector has an impact on the performance of this system is investigated as well.
Lastly, based on univariate proportional reversed hazard rate frailty model, we have a further study on two kinds of frailty models, one is the univariate general frailty model in terms of the reversed hazard rate, the other is the multivariate proportional reversed hazard rate frailty model. For the former, some monotone properties of the overall pop-ulation (mixture) reversed hazard rate and the population inactivity time are discussed. Besides, a sufficient condition for the preservation of DRHR under the mixture is derived as well. For the latter, we investigate how some of the well known stochastic orders of the frailty vector and some ageing properties of the baseline are translated into those of the overall population. Also, the stochastic monotone properties of the overall population vector with respect to the frailty vector are discussed. And the overall population reversed hazard rate is discussed as well.
Order statistics from heterogeneous observations play an important role in various areas related to applied probability and statistics such as reliability theory, survival anal-ysis, operations research and actuarial sciences. In the second part, we conduct stochastic comparisons on order statistics from independent heterogeneous exponential random vari-ables.
Firstly, we establish equivalent characterizations in terms of parameters for the right spread ordering between the second order statistic (Fail-Safe system) from nonidentical independent exponential observations and that from i.i.d. exponential ones.
Secondly, we build a sufficient condition for the hazard rate ordering between lifetimes of parallel systems with two independent but nonidentical exponential components, this also serves as an answer to the open problem of Joo and Mi (2010). Moreover, this result is extended to the case with proportional hazard rates. Some comparisons on lifetimes of such systems with general components are also obtained.
引文
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