The exact region determined by Kendall's tau, Spearman's footrule and Blomqvist's beta

Abstract

We determine the exact region ${\Omega }_{\tau ,\varphi ,\beta }:=\{(\tau (C),\varphi (C),\beta (C)):C\in \mathcal{C}\}$ of possible joint values of Kendall's tau, Spearman's footrule and Blomqvist's beta over the class $\mathcal{C}$ of all bivariate copulas. The region consists precisely of all triples $(t,p,b)$ satisfying $-1\le b\le 1$, $\frac{3}{16}(1+b{)}^2-\frac{1}{2}\le p\le 1-\frac{3}{8}(1-b{)}^2$ and $\frac{4}{3}p-\frac{1}{3}\le t\le \frac{2}{3}p+\frac{1}{3}$. In other words, the known exact $(\varphi ,\beta )$- and $(\tau ,\varphi )$-regions already characterize the joint region, so that, once the value of Spearman's footrule is fixed, Blomqvist's beta imposes no additional sharp restriction on the possible values of Kendall's tau. The proof is constructive: two one-parameter families of shuffles of $M$ realize the extreme values of Kendall's tau along the lower boundary of the $(\varphi ,\beta )$-region, ordinal sums spread these families through the whole region, and the vertical fibres are filled using the biaffinity of the concordance function. We further show that ${\Omega }_{\tau ,\varphi ,\beta }$ is convex with rectangular fixed-footrule sections, identify an affine symmetry of its fibres about $\tau =\varphi$, and compute its volume, which equals $\frac{31}{40}$.