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.

The concept of mean inactivity time plays an important role in reliability and life testing. In this investigation, based on the comparison of mean inactivity times of a certain function of two lifetime random variables, we introduce and study a new stochastic order. This new order lies between the reversed hazard rate and the mean inactivity time orders. Several characterizations and preservation properties of the new order under reliability operations of monotone transformation, mixture, and shock models are discussed. In addition, a new class of life distributions called strong increasing mean inactivity time is proposed, and some of its reliability properties are investigated. Finally, to illustrate the concepts, some applications in the context of reliability theory are included.

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.

The purpose of this paper is to enhance the study of the mean time to failure (MTTF) order in an age replacement model, and the decreasing mean time to failure (DMTTF) class of life distributions. We present new characterizations of the MTTF order in terms of the well-known hazard rate and reversed hazard rate orders. Preservation properties of this order under some reliability operations such as monotonic transformations, mixture, and weighted distributions are investigated. In this context, we provide new results related to the DMTTF class. Finally, the problem of testing exponentiality against such class is discussed, and some numerical results are presented.