On Kendall's Tau for Order Statistics

Abstract

Every copula $C$ for a random vector $\mathbf{X}=(X_1,\dots ,X_d)$ with identically distributed coordinates determines a unique copula $C_{:d}$ for its order statistic ${\mathbf{X}}_{:d}=(X_{1:d},\dots ,X_{d:d})$. In the present paper we study the dependence structure of $C_{:d}$ via Kendall's tau, denoted by $\kappa$. As a general result, we show that $\kappa [C_{:d}]$ is at least as large as $\kappa [C]$. For the product copula $\Pi$, which corresponds to the case of independent coordinates of $\mathbf{X}$, we provide an explicit formula for $\kappa [{\Pi }_{:d}]$ showing that the inequality between $\kappa [\Pi ]$ and $\kappa [{\Pi }_{:d}]$ is strict. We also compute Kendall's tau for certain multivariate margins of ${\Pi }_{:d}$ corresponding to the lower or upper coordinates of ${\mathbf{X}}_{:d}$.