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

Abstract

Let $X_1,X_2,\dots ,X_n$ (resp. $Y_1,Y_2,\dots ,Y_n$) 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. ${\delta }_i$), $i=1,2,\dots ,n$. Here it is shown that if $\left({\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right)$ is $p$-larger than (resp. weakly supermajorizes) $\left({\delta }_1,{\delta }_2,\dots ,{\delta }_n\right)$, 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, $i=1\dots n$, 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.

Due to some mathematical issues, we want to withdraw this article