由Spearman's rho和Gini's gamma确定的精确区域
The exact region determined by Spearman's rho and Gini's gamma
- Paris Lodron Universität Salzburg（萨尔茨堡大学）
- University of Freiburg（弗赖堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文确定了Spearman's rho和Gini's gamma在所有二元copula上的精确可达区域，通过显式参数化边界并构造极值copula，解决了该成对精确区域问题，并给出了二者差异的最大值及同号阈值。
AI中文摘要：
我们确定了所有二元copula上Spearman's rho和Gini's gamma的精确可达区域，解决了Spearman's rho、Kendall's tau、Gini's gamma、Blomqvist's beta和Spearman's footrule之间剩余的成对精确区域问题。我们给出了在给定Gini's gamma下rho最大边界的显式参数化，并构造了达到每个边界点的copula。该边界由一个基本弧段组成，直至一个单一连接点，在连接点之后，由可数个代数片段组成，这些片段累积于共单调性。这一描述还给出了$|\rho-\gamma|$的最大可能值，以及超过该阈值后rho和gamma必须同号的尖锐阈值。我们的证明将中心秩的符号和大小分离，将$\rho$和$\gamma$的线性组合的优化简化为两个均匀大小的最优传输问题。极值点结合了一个中心反对角块与重新缩放的辅助传输优化器，全局最优性由通过粘合相应对偶片段获得的Kantorovich对偶势函数保证。
英文摘要：
We determine the exact attainable region of Spearman's rho and Gini's gamma over all bivariate copulas, resolving the remaining pairwise exact-region problem among Spearman's rho, Kendall's tau, Gini's gamma, Blomqvist's beta and Spearman's footrule. We give an explicit parametrization of the rho-maximal boundary for prescribed Gini's gamma and construct copulas attaining every boundary point. The boundary consists of an elementary arc up to a single junction and, beyond it, countably many algebraic pieces accumulating at comonotonicity. This description also gives the largest possible value of $|ρ-γ|$ and the sharp thresholds beyond which rho and gamma must have the same sign. Our proof separates the signs and magnitudes of centred ranks, reducing the optimization of linear combinations of $ρ$ and $γ$ to an optimal transport problem for two uniform magnitudes. The extremizers combine a central antidiagonal block with rescaled auxiliary transport optimizers, and global optimality follows from a Kantorovich dual potential obtained by gluing the corresponding dual pieces.
