On the median residual lifetime and its aging properties: A characterization theorem and applications

This paper is devoted to study several aspects of the median residual life function (MERLF). In reliability studies, it is well known that, although the MERLF has several advantages over the mean residual life function (MRLF), the MRLF has the good property of uniquely determining a life distribution whereas either the median residual life function (MERLF) or an α-percentile residual life does not have such good property. We shall give a characterization result where knowledge of both the MERLF and the survival function on an interval does uniquely determine the distribution. Moreover, in order to apply this characterization in practical situations, we propose a method to estimate the necessary information of the survival function. Relationships between analytical properties of the survival function and its associated MERLF are also obtained. Bryson and Siddiqui proved relationships among seven criteria for aging, out of which two contained the MRLF (decreasing MRLF and net decreasing MRLF). In this paper, we prove that the same pattern of relationships holds if the MRLF is replaced by the MERLF. We also examine the aging criteria corresponding to an increasing MERLF and show that there is no relation between the behavior (increasing or decreasing) of the MERLF and of the MRLF.