Ordering results between extreme order statistics in models with dependence defined by Archimedean [survival] copulas

Motivated by recent works about stochastic comparisons between extreme order statistics arising from heterogeneous and dependent random variables, where the dependency structure is defined by the family of Archimedean copulas and the marginal distributions follow some specific parametric distributions, in this work, we investigate the case in which the marginal distributions can have arbitrary distribution functions depending on some parameter. Such parameter can be a shape, scale or location parameter, but other kinds of parameters, as frailty, resilience or tilt parameters can be also considered. Hence, the modified proportional hazard rate scale (MPHRS) and the modified proportional reversed hazard rate scale (MPRHRS) models, among others, belong to the wide parametric model studied here. Under this setup, we provide some general results for the usual stochastic order, when the parameter vectors verify the p-larger order or the reciprocally majorization order, generalizing some of the existing results in the literature. Besides this, extreme order statistics arising from the dependent MPHRS and MPRHRS models are compared in the sense of the reversed hazard rate order and the hazard rate order as well.