Joint distribution of maxima of concomitants of subsets of order statistics

Abstract

Let (X_i:n, Y_[i:n]), 1≤i≤n, denote the n pairs obtained by ordering a random sample of size n from an absolutely continuous bivariate population on the basis of X sample values. Here Y_[i:n] is called the concomitant of the ith order statistic. For 1≤k≤n, let V₁=max{{Y_[n-k+1:n],...,Y_[n:n]} and V₂=max{Y_[1:n],...,Y_[n-k:n]}. In this paper, we discuss the finite-sample and asymptotic joint distribution of (V₁,V₂). The asymptotic results are obtained when k=[np], 0<p<1, and when k is held fixed, as n→∞. We apply our results to the bivariate normal population and indicate how they can be used to determine k such that V₁ is close to Y_n:n, the maximum of the values of Y in the sample.