Ordering Properties of Order Statistics from Heterogeneous Generalized Exponential and Gamma Populations (original) (raw)

Let X1,X2,ldots,XnX_1, X_2,\ldots, X_nX1,X2,ldots,Xn (resp. Y1,Y2,ldots,YnY_1, Y_2,\ldots, Y_nY1,Y2,ldots,Yn) be independent random variables such that XiX_iXi (resp. YiY_iYi) follows generalized exponential distribution with shape parameter thetai\theta_ithetai and scale parameter lambdai\lambda_ilambdai (resp. deltai\delta_ideltai), i=1,2,ldots,ni=1,2,\ldots, ni=1,2,ldots,n. Here it is shown that if left(lambda1,lambda2,ldots,lambdanright)\left(\lambda_1, \lambda_2,\ldots,\lambda_n\right)left(lambda1,lambda2,ldots,lambdanright) is ppp-larger than (resp. weakly supermajorizes) left(delta1,delta2,ldots,deltanright)\left(\delta_1,\delta_2,\ldots,\delta_n\right)left(delta1,delta2,ldots,deltanright), then Xn:nX_{n:n}Xn:n will be greater than Yn:nY_{n:n}Yn:n in usual stochastic order (resp. reversed hazard rate order). That no relation exists between Xn:nX_{n:n}Xn:n and Yn:nY_{n:n}Yn:n, under same condition, in terms of likelihood ratio ordering has also been shown. It is also shown that, if YiY_iYi follows generalized exponential distribution with parameters left(barlambda,thetairight)\left(\bar\lambda,\theta_i\right)left(barlambda,thetairight), where barlambda\bar\lambdabarlambda is the mean of all lambdai\lambda_ilambdai's, i=1ldotsni=1\ldots ni=1ldotsn, then Xn:nX_{n:n}Xn:n is greater than Yn:nY_{n:n}Yn: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.