Distribution of extremal order statistics from large subsets of concomitants

From a random sample of size n from an absolutely continuous bivariate population (X,Y) we consider two complementary (upper and lower) subsets of Y-values formed by a sorting on the basis of the corresponding X-values. We derive the finite-sample and asymptotic joint distributions of the extreme order statistics of these Y subsets assuming that the subset sizes remain proportional to n as n→∞. We illustrate the use of our results with the bivariate normal example and provide an approximation to the probability of an event of interest in selection problems.