On some stochastic comparisons of arithmetic and geometric mixture models

Most studies on reliability analysis have been conducted in homogeneous populations. However, homogeneous populations can rarely be found in the real world. Populations with specific components, such as lifetime, are usually heterogeneous. When populations are heterogeneous, it raises the question of whether these different modeling analysis strategies might be appropriate and which one of them should be preferred. In this paper, we provide mixture models, which have usually been effective tools for modeling heterogeneity in populations. Specifically, we carry out a stochastic comparison of two arithmetic (finite) mixture models using the majorization concept in the sense of the usual stochastic order, the hazard rate order, the reversed hazard rate order and the dispersive order both for a general case and for some semiparametric families of distributions. Moreover, we obtain sufficient conditions to compare two geometric mixture models. To illustrate the theoretical findings, some relevant examples and counterexamples are presented.