Some stochastic comparison results of the highest and lowest order statistics of independent exponentiated Gumbel type-II distribution

Abstract This paper studies stochastic comparisons of the highest and lowest order statistics arising from independent Exponentiated Gumbel Type-II distributed random variables. First, using vector majorization, we establish a comprehensive set of stochastic orderings for the lowest and highest order statistics with respect to the outer and inner shape parameters. In particular, for the lowest order statistic, we derive results on the usual stochastic order and likelihood ratio order, which further imply failure rate, mean residual life, and increasing convex orderings. Second, we show that the lowest order statistics satisfy a proportional failure rate structure. In addition, we also establish star and dispersive orderings. For the highest order statistic, we obtain results in the reversed failure rate order, and we show that, in general, a proportional failure rate structure does not hold. Moreover, unlike several classical exponentiated models, the highest order statistics are not necessarily comparable in the usual stochastic order under heterogeneity of the inner shape parameters. Finally, we extend the analysis using multivariate chain majorization to study the effect of simultaneous variation in scale and outer shape parameters. Under this framework, we establish usual stochastic ordering results for both the lowest and highest order statistics. It provides more general reliability comparisons under joint parameter heterogeneity.