AI 中文总结
该研究针对统计学中Kendall's tau与Spearman's rho可能不一致的问题，证明了一类由分段线性单调满射生成的可分解Copula满足τ_C=ρ_C，并推导了其Chatterjee秩相关系数等相依测度的相关性质。
AI 中文摘要
Kendall's tau与Spearman's rho是统计学和风险管理中广泛使用的两种相依测度，常被视为可互换，但二者可能存在显著差异：Schreyer等人（2017）确定了可实现(τ,ρ)对的精确区域。我们研究互补问题——等式，即识别满足τ_C=ρ_C的非平凡Copula族。我们证明，对于形如C_{e,α}∗C_{β,e}的每个可分解Copula，该等式均成立，其中α和β为分段线性单调满射（PLMS）。针对这些PLMS生成的Copula，我们进一步研究包括Chatterjee秩相关系数和尾部相依系数在内的其他相依测度，揭示了一些有用的代数公式和意外现象，特别是Chatterjee系数可表现出极端不对称性。
英文摘要
Kendall's tau and Spearman's rho, two widely used dependence measures in statistics and risk management, are often treated as interchangeable, yet can disagree sharply: Schreyer et al.~(2017) established the exact region of attainable $(τ,ρ)$ pairs. We study the complementary question of equality, namely, identifying nontrivial families of copulas $C$ satisfying $τ_C=ρ_C$. We prove that this equality holds for every factorizable copula of the form $C_{e,α}\ast C_{β,e}$, where $α$ and $β$ are piecewise linear monotonic surjections (PLMS). For these PLMS-generated copulas, we study further dependence measures, including Chatterjee's rank correlation coefficient and tail dependence coefficients, revealing some useful algebraic formulas and unexpected phenomena. In particular, Chatterjee's coefficient can exhibit extreme asymmetry.
