On the exact region between Chatterjee's rank correlation and Spearman's footrule

Abstract: Chatterjee's rank correlation (\xi) has emerged as a popular measure quantifying the strength of directed functional dependence between random variables $X$ and $Y$. If $X$ and $Y$ are continuous, $\xi$ equals Spearman's footrule~(\psi) for the Markov product of the copula induced by $(X,Y)$ and its transpose. We analyze the relationship between these two measures more in depth by studying the attainable region of possible pairs ((\xi, \psi)) over all bivariate copulas. In particular, we show that for given $\xi$, the maximal possible value of $\psi$ is uniquely attained by a Fr\'echet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that (\xi\le\psi\le \sqrt{\xi}) characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of (\psi) for given (\xi), we give a lower bound based on Jensen's inequality and construct a two-parameter copula family that comes comparably close.