The exact region between Chatterjee's and Blest's rank correlations

TL;DR

This paper precisely characterizes the set of all possible pairs of Chatterjee's and Blest's rank correlations over all bivariate copulas, introducing a new extremal copula family and deriving explicit formulas for the boundary.

Contribution

It determines the exact joint region between Chatterjee's and Blest's rank correlations, introducing a novel extremal copula family and explicit boundary formulas.

Findings

Explicit boundary of the correlation region derived

A new extremal copula family identified

Closed-form expressions for correlations obtained

Abstract

Exact regions between rank correlations describe the set of all pairs of values that two dependence measures can attain simultaneously on the same copula and thus yield sharp inequalities between them. In this paper, we determine the exact region between Chatterjee's rank correlation $\xi$ and Blest's rank correlation $\nu$ over the class of all bivariate copulas. Our approach is based on a constrained optimization problem whose solution is characterized by Karush--Kuhn--Tucker conditions. This leads to a novel extremal copula family that uniquely traces the boundary of the region. For this family, we derive closed-form expressions for both $\xi$ and $\nu$, which provide an explicit parametrization of the exact attainable region.