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

TL;DR

This paper explores the relationship between Chatterjee's rank correlation and Spearman's rho, characterizing their attainable pairs and establishing bounds and inequalities for different dependence structures.

Contribution

It introduces the ta-0-region, a convex set describing possible pairs of these correlations, and proves bounds relating ta and 0 for various dependence scenarios.

Findings

The ta-0-region is convex with a boundary characterized by novel copulas.

ta always less than or equal to |0| when Y is stochastically monotone.

The maximum difference between ta and 0 is exactly 0.4.

Abstract

The rank correlation \xi(X,Y), recently established by Sourav Chatterjee 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 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 study the attainable set of pairs (\xi(X,Y),\rho(X,Y)). The resulting {\xi}-\r{ho}-region is a convex set whose boundary is characterized by a novel family of absolutely continuous, asymmetric copulas having a diagonal band structure. Moreover, we prove that \xi(X,Y)\leq|\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 ordering properties for \xi and \rho. Our results contribute to a better understanding of Chatterjee's rank correlation, which typically yields substantially smaller values than Spearman's \rho when quantifying positive dependencies. In particular, when interpreting the values of Chatterjee's rank correlation on the scale of \rho, the quantity \sqrt{\xi} appears to be more appropriate.