Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables

Abstract

In this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let X1,…,Xn be independent random variables with Xi∼EW(αi,βi,γi), i=1,…,n. Further, let X1∗,…,Xn∗ be another set of independent random variables with Xi∗∼EW(αi∗,βi∗,γi∗), i=1,…,n. First, when γ1=⋯=γn=γ1∗=⋯=γn∗ and a matrix with different parameters αi,βi changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when α1=⋯=αn=α1∗=⋯=αn∗,β1=⋯=βn=β1∗=⋯=βn∗ and (γ1,…,γn)⪰m(γ1∗,…,γn∗), we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.