The Asymptotic Property And Total Positivity Of Order 2 Of Bivariate Order Statistics

This paper is intended to make a study on the properties of bivariate order statistics (Xr,n,Ys,n).On the one hand, this paper discusses the convergence of (Xr,n,Ys,n):Let (X1,Y1),(X2,Y2),be i.i.d with marginal F(x) and G(y), and U1,U2～C(0,1),then, as nâ†'∞, (X[nU1],n,Y[nU2],n) converges almost surely to (F-1(U1),G-1(U2)). For any x and y, we have Particularly, if (U1,U2) and (X,Y1),(X2,Y2),...are independent, then By applying this conclusion, we can generalize the theorem 5 of [16], prove this theorem and provide the speed of convergence without " H(x,y) is PQD". Moreover, this paper simulates the tendency of error between (X[nU1],n,Y[nU2],n)’s empirical distribution function and P(F-1(U1)≤x,G-1(U2)≤y), as nâ†'∞, in R. In addition, for any positive integer n, has the same distribution as (F-1(U1),G-1(U2)).On the other hand, this paper also investigates the dependence structures TP2:If the joint distribution function of multivariate random variable (Z,, Z2, …, Zn) is MTP2, then the joint distribution function (ξ1,ξ2,…,ξm) is MTP2 where ξi= maxk∈TiZk, for i= 1,2,…m. If the joint survival function of multivariate random variable (Z,,Z2,…,Zn) is MTP2, then the survival distribution function (ξ1,ξ2,…,ξm) is MTP2 where ξi=mink∈TiZk, for i= 1,2,…m. Moreover, this article analyses the properties of TP2 and RR2 of min(F(x),G(y)), max(0,F(x)+G(y)-1) and F(x)G(y). In addition, it not only points the fact that Marshall and Olkin bivariate exponential family’s survival function and FGM model’s distribution function is TP2, but also calculates two specific distribution functions.