Stochastic Comparisons Of Order Statistics From Heterogeneous RS And Multiple-outlier PHR Samples

By adding a resilience parameter to the scale model,a general distribution family called resilience-scale model is introduced,which includes exponential,Weibull,gen-eralized exponential,exponentiated Weibull and exponentiated Lomax distributions as special cases.In this paper,we first carry out stochastic comparisons on parallel and series systems made up of heterogeneous resilience-scaled(RS)components.On the one hand,it is shown that more heterogeneity among the RS components of a paral-lel(series)system with an Archimedean(survival)copula will lead to its better(worse)performance in the sense of the usual stochastic order.On the other hand,the(reversed)hazard rate order is established for two series(parallel)systems comprised of indepen-dent heterogeneous RS components.Second we investigate the ordering properties of the second-order statistics arising from two sets of independent multiple-outlier propor-tional hazard rates(PHR)samples.It is proved that the submajorization order between the sample size vectors together with the supermajorization order between the hazard rate vectors imply the hazard rate ordering between the corresponding second-order s-tatistics from multiple-outlier PHR random variables.Last,some numerical examples and practical applications in system reliability and auction theory are provided to expli-cate the theoretical results.These results not only generalize and extend some of those known in the literature,but also provide theoretical guidance both for system design in reliability and the winner's price for the bid in the second-price reverse auction in auction theory.