Abstract
Let (Xk,Yk), k=1,2,…,n, be independent copies of bivariate random vector (X,Y) with joint cumulative distribution function F(x,y) and probability density function f(x,y). For 1≤r,s≤n, the vector of order statistics of X1:n≤X2:n≤⋯≤Xn:n and Y1:n≤Y2:n≤⋯≤Yn:n, respectively, is denoted by (Xr:n,Ys:n). Let (Xn+i,Yn+i), i=1,2,…,m, be a new sample from F(x,y), which is independent from (Xk,Yk), k=1,2,…,n. Let ξ1 be the rank of order statistics Xr:n in a new sample Xn+1,Xn+2,…,Xn+m and ξ2 be the rank of order statistics Ys:n in a new sample Yn+1,Yn+2,…,Yn+m. We derive the joint distribution of discrete random vector (ξ1,ξ2) and a general scheme wherein the distributions of new and old samples are different is considered. Numerical examples for given well-known distribution are also provided.