The exact region between Chatterjee's $\xi$ and Blomqvist's $\beta$

Abstract

We determine the exact attainable region of the pair $(\xi (C),\beta (C))$ formed by Chatterjee's rank correlation $\xi$ and Blomqvist's $\beta$ over the class of all bivariate copulas and show that it is given by $\{(x,y)\in [0,1]\times [-1,1]:|y{|}^3\le 2x\}.$ The left boundary $\xi =|\beta {|}^3/2$ is attained by an explicit two-strip family $(L_b{)}_{b\in [-1,1]}$ obtained by perturbing independence with a signed tent function $g_b$ centered at the median. We derive several properties of this copula family including the formulas for its density and rank correlation measures, as well as positive and negative dependence properties. The right boundary $\xi =1$ is attained for every admissible value of $\beta$ by deterministic measure-preserving copulas, and the full region is obtained by taking convex mixtures of the left- and right-boundary copulas with fixed $\beta$ and using the continuity of $\xi$ along these mixtures. We also record the exact regions in several natural subclasses of copulas.