Characterizations of the RHR and MIT orderings and the DRHR and IMIT classes of life distributions

dTwo 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-out-of-n system given that the time of the (n - k + 1) st failure occurs at or sometimes before time t >= 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.