The purpose of this article is to study several preservation properties of stochastic comparisons based on the mean inactivity time order under the reliability operations of convolution and mixture+ Characterizations and relationships with the other well-known orders are given+ Some examples of interest in reliability theory are also presented+ Finally, testing in the increasing mean inactivity time class is discussed+

The purpose of this article is to study several preservation properties of the mean inactivity time order under the reliability operations of convolution, mixture, and shock models. In that context, the increasing mean inactivity time class of lifetime distributions is characterized by means of right spread order and increasing convex order. Some applications in reliability theory are described. Finally, a new test of such a class is discussed.

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In this paper, relative behavior of two frailty models with different frailty variables and a common baseline variable are considered. We show that, under some conditions on frailty variables, the ratio of the hazard rate (mean residual life) functions of two frailty models is increasing. Finally, some applications are examined to validate the obtained results.

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Two well-known orders that have been introduced and studied in reliability theory are defined via stochastic comparison of inactivity time: the reversed hazard rate order and the mean inactivity time order+ In this article, some characterization results of those orders are given+ We prove that, under suitable conditions, the reversed hazard rate order is equivalent to the mean inactivity time order+ We also provide new characterizations of the decreasing reversed hazard rate~increasing mean inactivity time! classes based on variability orderings of the inactivity time of k-outof-n system given that the time of the~n Ϫ k ϩ 1!st failure occurs at or sometimes before time t Ն 0+ Similar conclusions based on the inactivity time of the component that fails first are presented as well+ Finally, some useful inequalities and relations for weighted distributions related to reversed hazard rate~mean inactivity time! functions are obtained+