On Kendall's Tau for Order Statistics

Every copula $C$ for a random vector ${\bf X}=(X_1,\dots,X_d)$ with identically distributed coordinates determines a unique copula $C_{:d}$ for its order statistic ${\bf X}_{:d}=(X_{1:d},\dots,X_{d:d})$. 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 ${\bf X}$, 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 ${\bf X}_{:d}$.