【作者】 杨建萍；

【导师】 胡太忠；

【作者基本信息】 中国科学技术大学 ， 统计学， 2016， 博士

【摘要】 本篇博士学位论文主要研究了随机变量的一类波动性度量、几种随机序与期望效用这三者之间的关系.主要内容包括如下的几个方面：1.对一类随机变量的波动性度量,研究该度量与常见随机序之间的关系.Lopez-Diaz et al.(2012)利用一个风险随机变量的生存函数与其扭曲生存函数之间的LP距离△h,p(X)来刻画风险变量X的波动性,并研究了当p≥1时该波动性度量在分散序下的保号性质,即若X在分散序意义下小于Y且p≥1,那么△h,p(X)不超过△h,p(Y).我们对p>0情形给出了此结果的一个简洁证明,并纠正了文献中的一个证明漏洞.考虑到分散序、位置独立风险序、剩余财富序、膨胀序以及凸序是比较常见的几种随机序,我们充分利用凸序的生成过程,寻找施加于扭曲函数h和参数p上的条件使得上述波动性度量的保号性质可以拓展到凸序、位置独立风险序、剩余财富序、膨胀序、洛伦茨序和星序情形.同时,我们给出了主要结果在次序统计量里的一个应用.2.基于期望效用理论给出常见寿命分布类的刻画,并将该型刻画用于建立几个随机序在卷积等运算下封闭性的研究Jewitt(1987,1989)研究了当效用函数单调增加而且是凹函数时,基于静态比较结果的封闭性给出反向失效率递减(DRHR)寿命分布类性质的刻画,并且进一步地得到了如果两个风险变量之间静态比较结果成立与位置无关,则这两个变量之间满足位置独立风险序.我们基于期望效用理论给出似然比单调增、失效率单调增失、广义似然比单调增、广义失效率单调增这四类寿命分布类的刻画,并将这些主要结果应用于建立分散序、位置独立风险序、剩余财富序、试验总时间变换序和星序在卷积或乘积运算下的封闭性.3.进一步研究两样本次序统计量的似然比序的比较.设X1,...,Xp是来自于总体分布为F,样本容量为p的一个样本；Xp+1,…,Xn是另一个来自总体分布为G,样本容量为q的样本,其中n=p+q并且0≤p<n.进一步假设两个样本是相互独立的.记两样本X1,...,Xn的次序统计量为X1:n(p,q)≤X2:n(p,q)≤…≤Xn:n(p,q)我们利用积和式理论证明了对任意k=1,…,n,若G在似然比序意义下小于F,则Xk:n(p,q)也在似然比序意义下小于Xk:n(p+1,g-1).本结果丰富了Zhao & Balakrishnan (2012),Ding et al.(2013)中的主要结果.更多还原

【Abstract】 This thesis is devoted to the investigation of the relations between one variability measure of a random variable, some stochastic orders, and expected utilities. The main results are as follows.1. The Lp-metric △h,p(X) between the survival function F of a random variable X and its distortion h o F is a characteristic of the variability of X. Lopez-Diaz et al. (2012) proved that if X is smaller than Y in the dispersive order, then △h,P(X)≤ △h,p(Y) for all increasing distortion functions h and p≥1. However, their proof is lengthy and there is one gap. In this paper, we gave an alternative and simple proof of the above result for p>0. We also proved that, under the assumptions that h is convex (or concave) and p E (0,1], △h,p(X)≤△h,p(Y) if X is smaller that Y in the sense of the location-independent risk order, the excess wealth order or the convex order. The corresponding results for some other stochastic orders and some applications of the main results are also presented.2. We characterized some ageing notions in terms of expected utilities, and then established the closure properties of some stochastic orders under the operation of con-volution or product. Aging notions were introduced as a tool in reliability theory, in many supply chain models and in stochastic models of applied probability. It is mean-ingful to study their characteristics. Under the assumptions that the utility functions are increasing and concave, Jewitt (1987,1989) characterized the ageing notion of DRHR (decreasing reversed hazard rate) and the location-independent risk order by using comparative statics. In this paper, we presented characterizations of the ageing notions of ILR (increasing likelihood ratio), IFR (increasing failure rate), IGLR (in-creasing generalized likelihood ratio) and IGFR (increasing generalized failure rate) in terms of expected utilities. Based on these characterizations, we investigated some conditions under which the dispersive order, the location-independent risk order, the excess wealth order, the total time on test transform order and the star order are closed under convolution or product.3. There is a large amount of literature on stochastic comparisons of order statis-tics and spacings. We established the stochastic comparisons of order statistics from two samples in the sense of the likelihood ratio order. Let X1,..., Xp be a random sample of size p from a distribution F, and Xp+1,..., Xn be another independent ran-dom sample of size q from a distribution G, where n=p+q,0≤p<n. Denote by Xk:n(p, q) the kth order statistic from two samples X1,...,Xn. It was shown that if G is smaller than F in the sense of likelihood ratio order, then Xk:n(p,q) is also smaller than Xk:n(p+1,q-1) in the sense of likelihood ratio order. The main result strengthens and complements some results in Zhao and Balakrishnan (2012) and Ding et al.(2013).更多还原