Some Results on Comparisons of Random Extremes having Identical and Non-Identical Components

TL;DR

This paper investigates the ordering of extremes in independent random variables, providing new conditions for hazard rate preservation and clarifying differences between identical and non-identical cases.

Contribution

It introduces new sufficient conditions for hazard rate orderings and highlights distinctions between identical and non-identical variable cases.

Findings

Conditions for hazard rate preservation are established.

Results hold for both identical and non-identical variables.

Non-identical case findings do not extend to identical cases.

Abstract

In this article, we revisit the paper by Kundu et al.~(2024), presenting new results and insights for both identical and non-identical independent random variables. We derive sufficient conditions for preserving the hazard rate and reversed hazard rate orderings between the random maximum and minimum order statistics, respectively. Our results show that these preservation conditions also hold for independent and identically distributed random variables. We also demonstrate that the findings in Kundu et al.~(2024) for the non-identical cases do not apply to the identical cases.