Joint distribution of new sample rank of bivariate order statistics

Let (X-k, Y-k), 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.1 be the rank of order statistics Xr: n in a new sample Xn+ 1, Xn+ 2,..., Xn+ m and.2 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 (.1,.2) 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.