Various Ordering Results between Two Finite Mixtures Arising from Some Families of Distributions

Abstract

Finite mixture models are useful for modeling heterogeneous data sets. It is mainly applicable for the reliability analysis of a system with multiple components. It also combines a finitely many subpopulations, when they have infinitely many elements. The stochastic comparison of two finite mixtures is an interesting topic which could be useful to compare two systems with multiple components, heterogeneous in nature. This thesis focuses on the study of various stochastic comparisons of finite mixture random variables. The main goal of the thesis is to establish several ordering results between two finite mixture random variables with respect to the usual stochastic order, hazard rate order, reversed hazard rate order, likelihood ratio order, ageing faster order in terms of reversed hazard rate, dispersive order, star order, Lorenz order, and right-spread order. When random variables of the subpopulations of a mixture model are chosen from general and specific distributions, such as the general parametric, location-scale, exponentiated location-scale, generalized Weibull, and inverted-Kumaraswamy, several sufficient conditions are proposed to compare two finite mixture random observations. In particular, we consider a general parametric family of distributions with cumulative distribution function FX (x) = F (x; ), x > 0, where > 0 is a model parameter. Here, the usual stochastic and hazard rate orders are established when the model parameter vectors are connected by p-larger and reciprocally majorization orders. Sufficient conditions are also obtained, under which the reversed hazard rate order holds between two mixture random variables. Further, the usual stochastic, hazard rate, reversed hazard rate, and dispersive orders are established between two mixture random variables, when a matrix of mixing proportions and model parameters changes to another matrix in a certain mathematical sense. Next, we consider the location-scale family of distributions with cumulative distribution function FX (x) = F (x ), x > , where > 0 and > 0 are the location and scale parameters, respectively. Here, we have established the usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order by taking heterogeneity in one parameter. Further, ordering results have been established by considering heterogeneity in two parameters with respect to the usual stochastic order and hazard rate order. We consider finite mixture models with subpopulations having exponentiated location-scale family of distributions with cumulative distribution function FX (x) = F (x ), x > , where > 0, > 0, and > 0 are the location, shape, and scale parameters, respectively. Here, we establish several ordering results in stochastic sense, and sufficient conditions are derived to compare two finite mixture models with respect to the usual stochastic order. The conditions depend on the weak supermajorization and reciprocally majorization orders. In order to validate some of the results, we have carried out a numerical simulation study. Next, we consider the generalized Weibull family of distributions with cumulative distribution function FX (x) = 1 e (!( x))β , x > 0, > 0, > 0, > 0, where !( x) = F ( x)/1 F ( x). Here, we present sufficient conditions under which the usual stochastic ordering, hazard rate ordering, and likelihood ratio ordering hold between the two finite mixture random variables. The sufficient conditions are based on the majorization and weak supermajorization orders between the associated model parameters. We also develop sufficient conditions based on unordered majorization order, under which two finite mixture models are comparable with respect to the usual stochastic order. We assume the inverted Kumaraswamy distribution with cumulative distribution function FX (x) = (1 (1 + x) ) , x > 0, > 0, > 0 as the components of the mixture random variables, and then find some ordering results between them. Here, we establish the usual stochastic order between two finite mixture random variables based on the concepts of weak supermajorization and weak submajorization orders. Also, we obtain comparison results with respect to the usual stochastic order and ageing faster order in terms of the reversed hazard rate when there is heterogeneity in two parameters. Further, we examine ordering results between the finite mixture models with respect to the reversed hazard rate and likelihood ratio orders. In addition, we have studied stochastic comparison results between two finite -mixture models with a general parametric family of distributions and generalized Weibull family of distributions in the sense of the usual stochastic ordering. In developing the usual stochastic order, we have proposed sufficient conditions which depend on the p-larger, reciprocally majorization, majorization, and weak supermajorization orders. In addition, we have also proposed sufficient conditions to compare two finite -mixture models based on the concept of unordered majorization order. we also derive sufficient conditions, under which the mixture random variables constructed by generalized Weibull family of distributions satisfy usual stochastic order for the case of -mixture models. Furthermore, we consider ordering results between two finite mixture models with multiple-outlier components. Here, we study two different problems: (i) finite arithmetic (classical) mixtures in multiple-outlier model with location-scale distributed components and (ii) finite -mixtures in multiple-outlier with general family distributed components, and the obtain some ordering results. In addition, multiple numerical examples and counterexamples are shown to demonstrate the efficacy of the established theoretical findings. Furthermore, a simulation study is carried out to estimate the model parameters of LSF of distributions in Chapter 9. Finally, the conclusion of the thesis with some new problems are presented.