Ordering of extreme order statistics among $q$-Weibull random variables

Abstract The $q$ -Weibull distribution, as a generalization of the Weibull distribution, plays an important role in the field of reliability theory, survival analysis, finance, engineering, medical science, etc. In contrast to the Weibull distribution, which is limited to describing monotonic hazard rate functions, the $q$ -Weibull distribution offers the flexibility to model various behaviors of the hazard rate function, including unimodal, bathtub-shaped, monotonic (both increasing and decreasing), and constant. In this article, we investigated the stochastic comparison of extreme order statistics derived from independent, heterogeneous $q$ -Weibull random variables using various stochastic orderings, including the usual stochastic order, hazard rate order, reversed hazard rate order, and likelihood ratio order. Some of these results are further extended to dependent setups by incorporating Archimedean copulas to model the dependence structure. Finally, we explored the behavior of extreme order statistics when the random variables are subjected to random shocks.