Exact attainable region for Chatterjee’s rank correlation and copula correlation ratio

Determine the exact attainable region \(\Omega_{\xi,\eta}\) of Chatterjee’s rank correlation \(\xi(X,Y)\) and the copula correlation ratio \(\eta(X,Y)\) over pairs of random variables with continuous marginal distribution functions, and establish whether either explicitly constructed inner-boundary function \(\eta_\ell\) or \(\eta_u\) is sharp.

Background

The paper studies the attainable region $\Omega_{\xi,\eta}$ relating Chatterjee’s rank correlation, which measures general functional dependence, to the copula correlation ratio, which measures the fraction of variance in the rank-transformed response explained by the predictor. By representing these quantities through Spearman’s footrule and Spearman’s rho of a Markov product, the paper derives explicit outer bounds for the region.

The authors also construct an inner enclosure bounded below by $\eta_\ell$ and above by $\eta_u$, with every point in the enclosure realized by a conditional-i.i.d. model. The exact region remains unknown because the Markov-product, or conditional-i.i.d., restriction is narrower than the class of all copulas; consequently, sharpness of the outer bounds for arbitrary copulas does not settle sharpness for $\Omega_{\xi,\eta}$.