On a conjecture about the comparability of parallel systems with respect to the convex transform order

Abstract

We study the comparability of the lifetimes of heterogeneous parallel systems with independent exponentially distributed components. It is known that the order statistics of systems composed of two types of components may be comparable with respect to the star transform order. On what concerns the stronger convex transform order, results have been obtained only for the sample maxima assuming that one of the systems is homogeneous. We prove, under the same assumptions as for the star transform ordering, that the lifetimes of heterogeneous parallel systems are not comparable with respect to the convex transform order.

Non-comparability with respect to the convex transform order with applications

“…However, as proved by Kochar and Xu [12] the ordering relationship holds with respect to a weaker form of transform order, namely the star transform order. Arab et al [5] provided a simpler proof of this latter ordering.…”

“…The specific order relation we will be considering was introduced by van Zwet [23] and is known either as the convex transform order (as in Shaked and Shanthikumar [21] or Kochar and Xu [11,12]) or as the increasing failure rate order (as in Averous and Meste [1], Fagiuoli and Pellerey [9], Nanda et al [16], Arab and Oliveira [2,3] or Arab et al [4,5]). As expressed by Definition 2 below, the convex transform order is defined through the relative convexity between the quantile functions.…”

“…However, the convex transform order may be adequately expressed through a control on the number of intersection points of the graphical representations of the distribution functions subject to the effect of zooming or shifting on the elapsing of time. This approach has been explored in Arab and Oliveira [2,3] and Arab et al [4,5] to derive explicit order relations within the Gamma and Weibull families of distributions and between these two families. Applications to ageing comparisons of parallel systems have been discussed by Kochar and Xu [11,12] and Arab et al [4,5].…”

“…This approach has been explored in Arab and Oliveira [2,3] and Arab et al [4,5] to derive explicit order relations within the Gamma and Weibull families of distributions and between these two families. Applications to ageing comparisons of parallel systems have been discussed by Kochar and Xu [11,12] and Arab et al [4,5]. The lifetime of parallel systems, described by the maximum of the lifetimes of each component, means that we need to consider distribution functions for which either it is not possible to obtain the inverse or this inverse is very difficult to be studied, even in the case of exponentially distributed components.…”

“…The comparison of parallel systems with exponentially distributed components, each one with a different hazard rate, was discussed in Kochar and Xu [11], conjecturing, based on numerical evidence, that the same convex transform order should hold under suitable relationships between the hazard rates of the components. The conjectured ordering was proved to be false by Arab et al [5], were it is shown that parallel systems with two components, with hazard rates satisfying the adequate relationship, are non-comparable by providing a general way to build counterexamples. However, as proved by Kochar and Xu [12] the ordering relationship holds with respect to a weaker form of transform order, namely the star transform order.…”