Distribution of concomitants of order statistics and their order statistics

For a random sample of size nn from an absolutely continuous random vector (X,Y)(X,Y), let Yi:nYi:n be iith YY-order statistic and Y[j:n]Y[j:n] be the YY-concomitant of Xj:nXj:n. We determine the joint pdf of Yi:nYi:n and Y[j:n]Y[j:n] for all i,j=1i,j=1 to nn, and establish some symmetry properties of the joint distribution for symmetric populations. We discuss the uses of the joint distribution in the computation of moments and probabilities of various ranks for Y[j:n]Y[j:n]. We also show how our results can be used to determine the expected cost of mismatch in broken bivariate samples and approximate the first two moments of the ratios of linear functions of Yi:nYi:n and Y[j:n]Y[j:n]. For the bivariate normal case, we compute the expectations of the product of Yi:nYi:n and Y[i:n]Y[i:n] for n=2n=2 to 8 for selected values of the correlation coefficient and illustrate their uses.