On some generalized ageing orderings

Abstract

Some partial orderings which compare probability distributions with the exponential distribution, are found to be very useful to understand the phenomenon of ageing. Here, we introduce some new generalized partial orderings which describe the same kind of characterization of some generalized ageing classes. We give some equivalent conditions for each of the orderings. Inter-relations among the generalized orderings have also been discussed.

Iterated Failure Rate Monotonicity and Ordering Relations Within Gamma and Weibull Distributions

“…We have been referring to the s−IFR as an ordering but, of course, one has to verify that this is really the case. This has been proved in [18]. Remark 3 This order relation is indeed only partial, as shown by the following example.…”

“…We will be only interested in the s−IFR ordering, so we do not quote here those other ordering notions. Moreover, as happens for the ageing notions, the s−IFR ordering is the strongest of those order relations, as proved in [18]. We have been referring to the s−IFR as an ordering but, of course, one has to verify that this is really the case.…”

“…In this section, we will define an iterated failure rate order and prove a general criterium. We start by defining the ordering, following Nanda, Hazra, Al-Mutairi and Ghitany [18].…”

“…It is possible to define several other ordering relations corresponding to the different ageing notions referred in Definition 4, as was done by Nanda, Hazra, Al-Mutairi and Ghitany [18]. We will be only interested in the s−IFR ordering, so we do not quote here those other ordering notions.…”

“…Theorem 13 (Theorem 2.2 in [18]) Let X be a random variable with distribution function F X ∈ F and Y with exponential distribution with scale parameter 1/λ. Then X ≤ s−IF R Y (resp., Y ≤ s−IF R X) if and only if X is s−IFR (resp., X is s−DFR).…”