The exact region and an inequality between Chatterjee's and Spearman's rank correlations

The rank correlation xi(X,Y), recently introduced by Chatterjee [12] and already popular in the statistics literature, takes values in [0,1], where 0 characterizes independence of X and Y, and 1 characterizes perfect dependence of Y on X. Unlike classical concordance measures such as Spearman's rho, which capture the degree of positive or negative dependence, xi quantifies the strength of functional dependence. In this paper, we determine, for each fixed value of xi, the full range of possible values of rho. The resulting xi-rho-region is a convex set whose boundary is characterized by a novel class of absolutely continuous, asymmetric copulas that are piecewise linear in the conditioning variable. Moreover, we prove that xi(X,Y) <= |rho(X,Y)| whenever Y is stochastically increasing or decreasing in X, and we identify the maximal difference rho(X,Y) - xi(X,Y) as exactly 0.4. Our proofs rely on a convex optimization problem under various equality and inequality constraints, as well as on a rearrangement-based dependence order underlying xi. Our results contribute to a better understanding of Chatterjee's rank correlation which typically yields substantially smaller values than Spearman's rho when quantifying the same positive dependence structure.