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

Chatterjee’s rank correlation ξ 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, ξ equals Spearman’s footrule ψ 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 ( ξ, ψ ) over all bivariate copulas. In particular, we show that for given ξ , the maximal possible value of ψ is uniquely attained by a Fréchet copula. As a by-product of this and a known result for Markov products of copulas, we obtain that ξ ≤ ψ ≤ ξ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of ψ for given ξ , we give a lower bound based on Jensen’s inequality and construct a two-parameter copula family that comes comparably close.