广义次序统计量的随机比较

【作者】 庄玮玮；

【导师】 胡太忠；

【作者基本信息】 中国科学技术大学 ， 概率论与数理统计， 2006， 博士

【摘要】 广义次序统计量是序贯次序统计量的一个子类，包含了许多概率统计中常用的有序变量的模型，例如，通常次序统计量、记录值、k-记录值、Pfeifer记录值、累进Ⅱ型删失次序统计量、多维不完全修理次序统计量，等等，广义次序统计量为我们提供了适当的方式来解释这些所含模型的相似性，通过将已有结果的推广和已有性质的综合，使得对这些模型结构的认识更加清楚，本文主要研究了广义次序统计量以及其间隔在几种常见随机序意义下的随机比较，得到了广义次序统计量的许多新的性质，这些性质可应用到其特殊的随机模型中。 对广义次序统计量的特殊情形—通常次序统计量，我们建立了其m阶间隔在似然比序和失效率序意义下的进一步随机比较。 对一般参数意义下的广义次序统计量研究了其一些进一步的一维随机比较。对建立在相同分布上的广义次序统计量，获得了其似然比序、失效率序、和反向失效率序；对建立在不同分布上的广义次序统计量，得到了其失效率序和分散序。 对来自同一样本的广义次序统计量的一个子类建立了其一般p阶间隔在似然比序和失效率序意义下的随机比较关系，并且得到了其一般p阶间隔对log-concave和log-convex性质的封闭性；对来自不同样本的广义次序统计量，建立了它们一般p阶间隔之间在似然比序意义下的随机比较以及它们正则化间隔之间在失效率序和分散序意义下的随机比较关系。 最后，我们建立了一样本序贯次序统计量在多维似然比序、多维失效率序和多维通常随机序意义下的随机比较关系，从而推导出其在一维似然比序和一维通常随机序意义下的进一步的结果；根据序贯次序统计量与非齐次纯生过程事件发生时刻向量之间的一一对应性质，给出了两样本序贯次序统计量在多维似然比序、多维失效率序和多维通常随机序下的随机比较；并且给出了主要结果在特殊的序贯次序统计量模型以及非齐次纯生过程中的应用。更多还原

【Abstract】 Generalized order statistics (GOS’s, for short) are a subclass of sequential order statistics, and contain a variety of models of ordered random variables often used in probability and statistics, e.g., ordinary order statistics, record values, k-record values, Pfeifer’s record model, progressive type II censored order statistics, order statistics under multivariate imperfect repair, and so on. GOS’s provide a suitable approach to explain these similarities and analogies in those models and to generalize related results. Through integration of known properties the structure of the embedded models becomes clearer. This thesis focuses on stochastic comparisons of GOS’s and its spacings in several stochastic orders. We have obtained many new properties of GOS’s, which can be used in other special models of ordered random variables.For the particular case of GOS’s - ordinary order statistics, we establish some further results on univariate stochastic comparisons for m-spacings in the likelihood ratio and the hazard rate orders.We investigate some further univariate stochastic comparisons of GOS’s under the more general assumptions on the parameters of GOS’s. For GOS’s based on the same distribution, we present the preservation properties of the likelihood ratio, hazard rate and reversed hazard rate orders; For GOS’s based on two different distributions, we obtain the preservation properties of the hazard rate and the dispersive orders.For GOS’s from one sample, we establish several stochastic comparisons of general p-spacings in the likelihood ratio and the hazard rate orders, and moreover, preservation properties of the log-convexity and logconcavity of p-spacings are given; For GOS’s from two samples, we establish the likelihood ratio ordering of general p-spacings and the hazard rate and the dispersive orderings of normalizing simple spacings.We finally establish some results on multivariate stochastic comparisons of one-sample sequential order statistics in the multivariate likelihood ratio, the hazard rate, and the usual stochastic orders, so we can conclude some further results on univariate stochastic comparisons of sequential order statistics in the likelihood ratio and the usual stochastic orders. Furthermore, based on the interrelationship between sequential order statistics and epoch times of nonhomogeneous pure birth (NHPB) processes, we obtain stochastic comparisons of two-sample sequential order statistics in the multivariate likelihood ratio, the hazard rate, and the usual stochastic orders. Applications in some special models of sequential order statistics and in the NHPB process are also presented.更多还原