Some Results Of Order Statistic

Order statistic is very useful in statistical inference. According to the distribution the population, we can easily derive the cumulative distribution function and probability density function of a single order statistic, which plays an important role in the application of the order statistics. Next, we will give some detailed results when the distribution of the population is continuous.Suppose that X1,X2,…,Xn is a random sample from a continuous population with pdf f(x) and cdf F(x); let Xl:n≤X2:n≤…< Xn:n be the order statistics obtained by arranging the preceding random sample in increasing order of magnitude, and the realization is xl:n≤x2:n≤…≤xn:n. Then the the density function of the kth statistic Xk:n is In particular, if X～U(0,1), then Xk:n～Beta(k, n-k+1). Furthermore, according to the above method, we can derive the joint density functions of k order statistics.Since, F(X)～U(0,1), we know that the uniform distribution order statistic is very important. Specifically, when F(x) is continuous, we haveIn recent years, stochastic comparisons of order statistics have been widely used in prob-ability and statistics, reliability analysis, queueing theory, life testing, operations research, survival analysis, actuarial science, investment decision, etc, so stochastic comparisons of order statistics have received great attention around the world. Because most of the actual date have fat tails in reliability analysis, life testing, survival analysis. At the same time, the Gamma distribution and Weibull distribution have fat tails too. So the studies of stochastic comparisons of order statistics of Gamma and Weibull is becoming more important.First, we give two important definitions an follows.Definition 1:Let X=(X1,X2,…,Xn),Y=(Y1,Y2,…,Yn)denote two real-valued vectors.LetX(1)≥…≥X(n),Y(1)≥…≥Y(n)be their ordered components.Then Y is called majorized by X,in symbols X≥Y,if for j=1,2,…,n-1,andΣi=1n X(i)=Σi=1n Y(i).Definition 2:Let X=(X1,X2,…,Xn)and Y=(Y1,Y2,…,Yn)be random vectors,X is said to be larger than Y in the stochastic order if P(X∈U)≥P(Y∈U),for all upper set U(A set U(?)Rn is called upper set if Y∈U whenever Y≥X and X∈U),denoted by X≥st Y.Then we will introduce the theorems of stochastic comparisons of order statistics for Gamma distribution.Li Hong Sun discussed stochastic comparisons of order statistics of in-dependent but not identically Gamma distribution under control and obtain two conclusions, see[26]. [9]extended those conclusions from Gamma distribution to generalized Gamma distribution gp,q(x)=p/Γ(q/p)xq-1e-xp,(x>0,p>0,q>0).When the shape parameters p,g satisfy certain conditions,we can obtain the following theorem:Theorem 1:Let X=(X1,…,Xn),Y=(Y1,…,Yn)be independent generalized gamma random vectors with common shape parameter p,q(0