Variability Measures, Stochastic Orders And Expected Utilities

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).