Ordering Results for Extended Weibull Distribution Based on Samples of Random Size

Abstract

Consider two sets of dependent N1 and N2 variables {X1,…,XN1} and {Y1,…,YN2}, where Xi∼EW(αi,λi, ki) and Yi∼EW(βi,μi,li), coupled with Archimedean (survival) copula having different generators, with N1 and N2 being two positive discrete random variables having their support as {1,2,…}, independently of Xi′s and Yi′s, respectively. In this article, we then establish sufficient conditions for the comparison of X1:N1 and Y1:N2, and XN1:N1 and YN2:N2 according to the usual stochastic, hazard rate, reversed hazard rate and dispersive orders. In particular, we extend the results established recently by Samanta et al. [Orderings of extremes among dependent extended Weibull random variables. Probab Eng Inf Sci. 2024;38(4):705–732. doi: 10.1017/S026996482400007X] for the case of fixed sample size to the case of random sample size. Furthermore, the results obtained here strengthen and generalize some of the results known in the literature.