Stochastic Comparisons Of Order Statistics From Heterogeneous Semi-parametric Samples

Order statistics play an important role in reliability engineering,auction theory,actuarial science,life testing and survival analysis.And stochastic comparison is a hot topic in reliability engineering.In real life,the heterogeneity and dependence of samples are universal.This paper devotes to investigating the ordering properties on order statistics from two sets of the heterogeneous Semi-parametric(SP)samples by using the tool of stochastic order.First,we obtain the reversed hazard rate order between the largest order statistics from the independent and heterogeneous Semi-parametric samples.Second,for heterogeneous Semi-parametric random variables with the same Archimedean(survival)copula,it is shown that,under some conditions,the usual stochastic order is established of the largest(smallest)order statistics by using the concept of vector supermajorization order and submajorization order and the second largest(smallest)order statistics have been compared in the sense of usual stochastic order.It is also illustrated that the results are the expansion of PHR model and Scale model in some cases.In the end,for heterogeneous Semi-parametric random variables with the same general(survival)copula,the largest(smallest)order statistics have been compared in the sense of usual stochastic order.The results of this paper strengthen and extend some of the results of the existing literature.In addition,we give some numerical examples accordingly to verify the validity of the results.