Stochastic Comparisons In Mixture Models

Stochastic orders are now a hot topic of research in many diverse areas of prob-ability and statistics. Mixture model has been widely used in reliability theory, survival analysis, risk theory and some other areas related to probability and statis-tics. This thesis studies the stochastic properties of mixture models and establishes stochastic comparison results between those random variables or random vectors arising in mixture models based upon various stochastic orders.Firstly, we focus on the analysis of univariate mixture models. Here, we study two special univariate mixture models. The first one is a mixed proportional excess hazard model. Dependence properties and ageing properties of mixed proportional excess hazard model are studied. Moreover, some stochastic comparisons results relating to those random variables of this model are also presented. The second one is a mixed scale change model. We examine the stochastic properties of the mixtures of scale change model and establish dependence between the overall population variable and the unobserved mixing variable. We present sufficient conditions to see how well known stochastic orderings between two mixing (baseline) variables translate into orderings between the corresponding population variables.Secondly, we consider a system of n components sharing a common random environment. Lifetimes of components of a system are usually dependent since com-ponents may share the common environment or share the same load. One way of modeling situations where the components with random lifetimes operate in a com-mon random environment is to use a multivariate mixture model. Here, we consider some special multivariate mixture models, including multivariate mixed proportional hazard model, multivariate mixed proportional reversed hazard rate model, multi-variate mixed proportional odd model, multivariate stress-strength model and warm standby system. Through doing stochastic comparisons, we aim to study how the random environment affects the number of functioning components in a reliability system and the performance of a k-out-of-n system.Lastly, we consider the general multivariate mixture model. For this model, we obtain the relationship between the conditional and the unconditional reliability measures such as the hazard gradient, the reversed hazard gradient, the multivariate mean residual life and the multivariate reversed mean residual life. We present some sufficient conditions under which random vectors and conditional random variables of the general multivariate mixt.ure model can be stochastically compared. In par-ticular, we consider weak monotone likelihood ratio order, the weak multivariate hazard rate order, weak multivariate reversed hazard rate order and usual multi-variate stochastic order. We also give some examples showing that some sufficient conditions can not be dropped.