Joint distribution of new sample rank of bivariate order statistics

Abstract

Let , be independent copies of bivariate random vector with joint cumulative distribution function and probability density function . For , the vector of order statistics of and , respectively, is denoted by . Let , , be a new sample from , which is independent from . Let be the rank of order statistics in a new sample and be the rank of order statistics in a new sample . We derive the joint distribution of discrete random vector 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.