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

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.