The Total Time On Test Transform Orders And Stochastic Comparison Of Order Statistics

Stochastic comparison is very useful in applied probability,statistics,reliability,actuary science,and other fields,and stochastic orders also play very important role.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 typeⅡcensored order statistics,order statistics under multivariate imperfect repair,and so on.This thesis focuses on some intrinsic properties of total time on test(ttt)orders,its dual orders and stochastic comparisons of GOS's and its special models.Firstly,we redefine the ttt order which enables one to compare unnecessarily nonnegative random variables.A new partial order(≤dttt)is also introduced,which can be regarded as the dual of the order≤ttt.Several new intrinsic properties of these two orders are investigated.In particular,we establish an interesting separation result on connections between the order≤ttt [≤dttt]and the excess wealth order[location independent riskier order],and obtain generating processes of the order≤ttt[≤dttt]in terms of mean-decreasing right[mean-decreasing left] stretches.Closure properties of the orders≤ttt and≤dttt under convolutions,and under minima or maxima are also presented.Secondly,for the particular case of GOS's-ordinary order statistics,on the one hand, we consider the problems of optimal allocation of a r-out-of-n system with respect to the usual stochastic order and of optimal allocation of a series system with respect to the reversed hazard rate order.It is shown that for each r,by balancing the allocation of active redundancies one can stochastically maximize the lifetime of the resulting r-out-of-n system, and that there exists an optimal allocation to optimize the reversed hazard rate function of the system for n=2 while such an optimal allocation does not exist for n＞2;On the other hand,the multivariate usual stochastic orderings are established for order statistics from independent (with possibly nonidentical distributions)random variable;and the multivariate likelihood ratio orderings of order statistics conditioned on both the right tail and the left tail are built.These results strengthen and generalize those conclusions in terms of the univariate likelihood ratio order.Finally,we examine the preservation properties of class of life distributions DRHR based on GOSs and investigate less restrictive conditions on the model parameters which enable one to establish the likelihood ratio and the usual stochastic orderings of conditional generalized order statistics from two samples.The main results strengthen and generalize the corresponding results established recently in the literature.