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

Source abstract

Blest's rank correlation $ν$ is a variant of Spearman's rho $ρ$ that weights the leading ranks of one variable more heavily, at the price that $ν$ is not symmetric in its arguments. We quantify both features by determining the exact region of $(ρ,ν)$ over all bivariate copulas, as well as that of $(η,ν)$, where $η$ is the symmetrized Blest coefficient of Genest and Plante. The latter region is a linear image of the set of all pairs $(ν(C),ν(C^\top))$ 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 $ρ$ or $η$, each corresponding extreme value of $ν$ is attained by exactly one copula, given in closed form and supported on finitely many line segments. Near countermonotonicity, the upper extremizers of the $(η,ν)$-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.