Stochastic Ordering Of Order Satistics From Power Function Distributions

The order is called stochastic order which studys the relationship between random variables. It has a very important practical significance to study stochastic order. In the real world, when we meet some random models that are very complex and difficult to be strictly deal with, stochastic order provides us with good approximation method,and get what we care about certain amount of approximate bounds (or one of them); In addition, the research of stochastic order is also used queuing theory, reliability theory, epidemiol-gy, economics, actuarial science and other fields.This paper discusses the Powrer function distribution of the usual order statistics random varibles problem under controlled conditions, that is when one of the control of parameters to meet one of conditions, we discuss the usual order statistics random varibl-es which obey the Power function distribution of order statistics.This article has mainly utilized the definition of the Schur concave function and the Schur convex function, the necessary and sufficient condition and its properties to prove the order statistics of the uaual stochastic order problem.In this paper,we have get two conclusions as follows:(1)letX= (X1,X2,L,Xn)and and Y=(Y1,Y2,L, Yn) be the distribution of Power which has the same parameters c,but has different parameters (?)=((?)1,(?)2,…,(?)n),μ=(μ1,μ2,…,μn),and X1, X2, L, Xn Y1,Y2,…,Yn are independent,if (?)>μ,then (?)c>0,X(1)≤Y(i) i=1,2,…,n.(2)let X=(X1 X2,L,Xn) and Y=(Y1,Y2,L,Yn) be the distribution of Power which has the same parameters (?),but has different parameters c=(c1,c2,L,cn) and c'=(c'1,c'2,L,c'n),and X1,X2,…,Xn, Y1,Y2,…,Yn are independent,if c>c',then (?)>0, X(1)<≤st Y(i) i=1,2,…,n-1;X(n)=st, Y(n).In reliability theory study, the (?) system holds an important position, and system reliability theory of (?) is equivalent to the problem of order statistics.