Distribution of order statistics from selected subsets of concomitants

For a random sample of size n from an absolutely continuous bivariate population (X, Y), let Xi:n be the i th X-order statistic and Y[i:n] be its concomitant. We study the joint distribution of (Vs:m, Wt:n−m), where Vs:m is the s th order statistic of the upper subset {Y[i:n], i=n−m+1,…,n}, and Wt:n−m is the t th order statistic of the lower subset {Y[j:n], j=1,…,n−m} of concomitants. When m = ⌈ np 0 ⌉ , s = ⌈ mp 1 ⌉ , and t = ⌈ ( n − m ) p 2 ⌉ , 0 < p i < 1 , i = 0 , 1 , 2 , and n → ∞ , we show that the joint distribution is asymptotically bivariate normal and establish the rate of convergence. We propose second order approximations to the joint and marginal distributions with significantly better performance for the bivariate normal and Farlie–Gumbel bivariate exponential parents, even for moderate sample sizes. We discuss implications of our findings to data-snooping and selection problems.