On the dependence structure of order statistics and concomitants of order statistics

Abstract Let (Xi,Yi),i=1,…,n, be n independent identically distributed pairs of rv's. If the Xi are ordered, one obtains both the order statistics X1:n≤⋯≤Xn:n and their concomitants Y[1:n],…,Y[n:n], where Y[r:n],r=1,…,n, are the Yi paired with Xr:n. Some results of Tukey (1958) on the dependence structure of the Xi:n are reviewed and extended. Conditions are given for the Y[i:n] to be associated or to be multivariate totally positive of order 2 (MTP2). It is also shown that the covariance of two order statistics can be negative if the Xi are sufficiently negatively dependent.