Stochastic Comparisons of Parallel Systems of Heterogeneous Generalized Exponential Components

Abstract

Let X1;X2; : : : ;Xn (resp. Y1; Y2; : : : ; Yn) be independent random variables such that Xi (resp. Yi) follows generalized exponential distribution with shape parameter i and scale parameter i (resp. i), i = 1; 2; : : : ; n. Here it is shown that if = ( 1; 2; : : : ; n) majorizes = ( 1; 2; : : : ; n) then Xn:n will be greater than Yn:n in reversed hazard rate ordering. That no relation exists between Xn:n and Yn:n, under same condition, in terms of likelihood ratio ordering has also been shown. It is also shown that, if Yi follows generalized exponential distribution with parameters ; i, where is the mean of all i's, i = 1 : : : n, then Xn:n is greater than Yn:n in likelihood ratio ordering. In this context, an error in Marshall, Olkin and Arnold [Inequalities: Theory of Majorization and Its applications (2011)] has been corrected, and some new results on majorization have been developed.