Rank weighting and asymmetry in Blest's rank correlation: two exact regions

Abstract

Blest's rank correlation $\nu$ is a variant of Spearman's rho $\rho$ that weights the leading ranks of one variable more heavily, at the price that $\nu$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(\rho,\nu)$ over all bivariate copulas, as well as that of $(\eta,\nu)$, where $\eta$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(\nu(C),\nu(C^\top))$ formed by a copula $C$ and its transpose. Consequently, Blest's coefficient differs from Spearman's rho by at most $1/4$, and interchanging the two variables changes it by at most $27/64$, improving on the bound $1/2$ implied by the first inequality. For every given value of $\rho$ or $\eta$, each corresponding extreme value of $\nu$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(\eta,\nu)$-region are supported on the graph of a function of the second coordinate, yet their conditional laws given the first coordinate carry two atoms. The proofs rest on a rearrangement inequality with equality case and on explicit Kantorovich potentials.