Abstract
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-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.