Stochastic Comparisons Of Extremum From Heterogeneous Exponential Family Samples

Exponential family distribution includes many continuous and discrete correlation distributions,such as Exponential distribution,Weibull distribution,Gamma distribution,Poisson distribution,Binomial distribution and Negative Binomial distribution.There are a lot of researches on non-homogeneous order statistics in the literature,but they mainly focus on special distributions such as exponential distribution and Weibull distribution.Considering that exponential family distribution contains many common distributions,the research on exponential family distribution is particularly important.In this study,we investigate the ordering properties of extremum arising from heterogeneous exponential family samples.First of all,it is proved that the majorization order or weak majorization order between the parameter vectors imply the usual stochastic ordering and reserved hazard rate ordering for the largest order statistics arising from two sets of independent heterogeneous exponential family variables,and the majorization order or weak majorization order between the parameter vectors imply the usual stochastic ordering and hazard rate ordering for the smallest order statistics arising from two sets of independent heterogeneous exponential family variables.Also,we show that if the condition of parameter is strengthened,the likelihood ratio ordering holds for the largest order statistics arising from two sets of independent heterogeneous exponential family variables.For the case of two parameters,it is proved that the majorization order between one parameter vectors together with the weak majorization order between another parameter vectors imply the usual stochastic ordering and reserved hazard rate ordering for the largest order statistics arising from two sets of independent heterogeneous exponential family variables,and the majorization order between one parameter vectors together with the weak majorization order between another parameter vectors imply the usual stochastic ordering and hazard rate ordering for the smallest order statistics arising from two sets of independent heterogeneous exponential family variables.The new results established here strengthen and generalize some known results in the literature,which is of great importance in theory.