Some ordering properties of highest and lowest order statistics with exponentiated Gumble type-II distributed components

TL;DR

This paper investigates the stochastic ordering properties of the highest and lowest order statistics derived from the exponentiated Gumble type-II distribution, providing theoretical comparisons and illustrative examples.

Contribution

It introduces new stochastic comparison results for order statistics of the exponentiated Gumble type-II distribution using various ordering techniques.

Findings

Stochastic comparisons are established for order statistics with different parameters.

Examples support the theoretical stochastic ordering results.

Some orderings, like the highest order statistic, are shown to be non-comparable.

Abstract

In this paper, we have studied the stochastic comparisons of the highest and lowest order statistics of exponentiated Gumble type-II distribution with three parameters. We have compared both the statistics by using three different stochastic ordering. First, we consider a system with different scale and outer shape parameters and then we study the usual stochastic ordering of the lowest and highest order statistics in the sense of multivariate chain majorization. In addition, we construct two examples to support our results. Second, by using the vector majorization technique, we study the usual stochastic ordering, the reversed failure rate ordering and the likelihood ratio ordering with respect to different outer shape parameters, next, by varying the inner shape parameter, we discuss the usual stochastic order of the lowest order statistics and we have shown that the highest order statistics are not comparable in the usual stochastic ordering by an example.