Orderings of extremes among dependent extended Weibull random variables

Abstract

In this work, we consider two sets of dependent variables $\{X_{1},\ldots,X_{n}\}$ and $\{Y_{1},\ldots,Y_{n}\}$, where $X_{i}\sim EW(\alpha_{i},\lambda_{i},k_{i})$ and $Y_{i}\sim EW(\beta_{i},\mu_{i},l_{i})$, for $i=1,\ldots, n$, which are coupled by Archimedean copulas having different generators. Also, let $N_{1}$ and $N_{2}$ be two non-negative integer-valued random variables, independent of $X_{i}'$s and $Y_{i}'$s, respectively. We then establish different inequalities between two extremes, namely, $X_{1:n}$ and $Y_{1:n}$ and $X_{n:n}$ and $Y_{n:n}$, in terms of the usual stochastic, star, Lorenz, hazard rate, reversed hazard rate and dispersive orders. We also establish some ordering results between $X_{1:{N_{1}}}$ and $Y_{1:{N_{2}}}$ and $X_{{N_{1}}:{N_{1}}}$ and $Y_{{N_{2}}:{N_{2}}}$ in terms of the usual stochastic order. Several examples and counterexamples are presented for illustrating all the results established here. Some of the results here extend the existing results of Barmalzan et al. (2020).