Kendall and Spearman bounds for Chatterjee's rank correlation under positive dependence

Abstract

We compare Chatterjee's rank correlation $\xi$ with Kendall's $\tau$ and Spearman's $\rho$ under positive-dependence assumptions on bivariate copulas. Our main technical contribution is a sharp order-violation bound for two stochastically ordered distribution functions. This local inequality controls each conditional order-violation probability appearing in Kendall's tau by the cross-rank variance functionals that determine Chatterjee's rank correlation. As a consequence, we prove the sharp Kendall bound $\xi (C)\le \tau (C)$ for every stochastically increasing copula $C$. The bound is best possible: ordinal sums of product copulas attain equality. We also prove that the weaker left-tail decreasing (LTD) and right-tail increasing (RTI) conditions jointly imply the Spearman bound $\xi (C)\le \rho (C)$, with equality if and only if $C$ is either the independence or comonotonicity copula. Finally, checkerboard examples show that LTD or RTI alone does not imply $\xi (C)\le \rho (C)$, that LTD and RTI together do not imply $\xi (C)\le \tau (C)$, and that both bounds are directional for $\xi$.