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

Chatterjee’s rank correlation has emerged as a popular measure quantifying the strength of<br>directed functional dependence between random variables and . If and are continuous,<br> equals Spearman’s footrule for the Markov product of the copula induced by (, ) and its<br>transpose. We analyze the relationship between these two measures more in depth by studying the attainable region of possible pairs (, ) over all bivariate copulas. In particular, we show that for given , the maximal possible value of 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 ≤ ≤ √ characterizes the exact region of stochastically increasing copulas. Regarding the minimal possible value of for given , we give a lower bound based on Jensen’s inequality and construct a two-parameter copula family that comes comparably close.