Relative Aging of Distributions

Abstract

We consider the reliability of one random variable relative to another, when the variables are continuous and take values in an interval [a, b). We give definitions and characterizations for the exponential property and the standard aging properties IFR, IFRA, and NBU. The exponential property defines an equivalence relation on the distributions, and then each of these aging properties defines a partial order on the distributions, modulo the exponential equivalence. We give a set of conditions that must be satisfied for a general aging property to define such a partial order and show that these conditions are not satisfied by the NBUE property. Several parametric families of distributions are considered.

Some properties and applications of cumulative Kullback–Leibler information

“…This fact introduces a notion of relative aging, in the sense that X is aging faster (slower) than Y if and only if the ratio of hazard rates λ X ( t )/ λ Y ( t ) is increasing (decreasing) in t . See also the contribution by Rowell and Siegrist on this topic.…”

“…For instance, we recall that X is said to be aging faster than Y if the random variable Z = T Y (X) is IFR, where T Y (t) is defined in Equation (10) (see Definition 1 of Sengupta and Deshpande [32]). This fact introduces a notion of relative aging, in the sense that X is aging faster (slower) than Y if and only if the ratio of hazard rates X (t)∕ Y (t) is increasing (decreasing) in t. See also the contribution by Rowell and Siegrist [33] on this topic.…”

“…where Figure 6 shows some plots of  KL (X s 1 , X s 2 ) for various choices of the involved parameters. From (33), it is easy to check that  KL (X s , X s ) = 0, s > 0. Moreover, the following limits hold:…”

Stochastic Comparisons of Nonhomogeneous Processes

“…Sengupta and Deshpande (1994) and Rowell and Siegrist (1998) have shown that (3.4)==> (3.6) (in fact, they treated (3.4) and (3.6) as notions of relative aging of two life distributions).…”

Some results on the relative ordering of two frailty models

“…In the recent decades, many authors have developed stochastic orders based on the concept of relative aging (cf. Sengupta and Deshpande 1994;Rowell and Siegrist 1998;Di Crescenzo 2000;Lai and Xie 2003;Bhattacharjee et al 2013;Rezaei et al 2015 and the references therein).…”