Ordering Properties of Order Statistics from Heterogeneous Generalized Exponential and Gamma Populations

TL;DR

This paper investigates the stochastic ordering of the maximum order statistics from heterogeneous generalized exponential and gamma distributions, establishing conditions under which these orderings hold and exploring related majorization and system comparison results.

Contribution

It introduces new stochastic ordering results for order statistics from generalized exponential distributions under majorization conditions and extends the theory of majorization with new results.

Findings

Order statistics are ordered in stochastic sense under p-larger and supermajorization conditions.

Likelihood ratio ordering does not always hold under the same conditions.

Comparison of gamma series systems using stochastic orders.

Abstract

Let $X_1, X_2,\ldots, X_n$ (resp. $Y_1, Y_2,\ldots, Y_n$) be independent random variables such that $X_i$ (resp. $Y_i$) follows generalized exponential distribution with shape parameter $\theta_i$ and scale parameter $\lambda_i$ (resp. $\delta_i$), $i=1,2,\ldots, n$. Here it is shown that if $\left(\lambda_1, \lambda_2,\ldots,\lambda_n\right)$ is $p$-larger than (resp. weakly supermajorizes) $\left(\delta_1,\delta_2,\ldots,\delta_n\right)$, then $X_{n:n}$ will be greater than $Y_{n:n}$ in usual stochastic order (resp. reversed hazard rate order). That no relation exists between $X_{n:n}$ and $Y_{n:n}$, under same condition, in terms of likelihood ratio ordering has also been shown. It is also shown that, if $Y_i$ follows generalized exponential distribution with parameters $\left(\bar\lambda,\theta_i\right)$, where $\bar\lambda$ is the mean of all $\lambda_i$'s, $i=1\ldots n$, then $X_{n:n}$ is greater than $Y_{n:n}$ in likelihood ratio ordering. Some new results on majorization have been developed which fill up some gap in the theory of majorization. Some results on multiple-outlier model are also discussed. In addition to this, we compare two series systems formed by gamma components with respect to different stochastic orders.