AI 中文总结
本文确定了Blest秩相关与Spearman's rho及对称化Blest系数之间的精确可行区域，量化了其加权和非对称性，并给出了达到极值的唯一copula的闭式表达。
AI 中文摘要
Blest秩相关 $\nu$ 是Spearman's rho $\rho$ 的一个变体，它对一个变量的领先秩赋予更大的权重，代价是 $\nu$ 在其自变量上不对称。我们通过确定所有二元copula上 $(\rho,\nu)$ 的精确区域以及 $(\eta,\nu)$ 的精确区域来量化这两个特征，其中 $\eta$ 是Genest和Plante的对称化Blest系数。后一个区域是由copula $C$ 及其转置 $C^\top$ 形成的所有对 $(\nu(C),\nu(C^\top))$ 的集合的线性像。因此，Blest系数与Spearman's rho的差异至多为 $1/4$，并且交换两个变量会使其变化至多 $27/64$，改进了由第一个不等式隐含的 $1/2$ 界限。对于每个给定的 $\rho$ 或 $\eta$ 值，$\nu$ 的每个相应极值都由恰好一个copula达到，该copula以闭式给出并支撑在有限多条线段上。在接近反单调性时，$(\eta,\nu)$ 区域的上极值函数支撑在第二个坐标的某个函数的图像上，然而给定第一个坐标的条件分布携带两个原子。证明依赖于一个带等式情形的不等式以及显式的Kantorovich势。
英文摘要
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.
Comments19 pages, 4 figures
