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

TL;DR

This paper develops criteria to determine when two random variables are not comparable under the convex transform order, and applies these to various distributions, disproving a conjecture about their comparability.

Contribution

It introduces simple tools for non-comparability in convex transform order and broadens understanding of distribution comparisons in stochastic orders.

Findings

Criteria for non-comparability of random variables

Application to exponential, Weibull, and Gamma distributions

Disproof of a conjecture on convex transform order comparability

Abstract

In the literature of stochastic orders, one rarely finds results that can be considered as criteria for the non-comparability of random variables. In this paper, we provide results that enable researchers to use simple tools to conclude that two random variables are not comparable with respect to the convex transform order. The criteria are applied to prove the non-comparability of parallel systems with components that are either exponential, Weibull or Gamma distributed, providing a negative answer for a conjecture about comparability with respect to the convex transform order in a much broader scope than its initial statement.