Tau-Rho Equality and Other Dependence Measures of a Subclass of Factorizable Copulas

Abstract

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 $(\tau ,\rho )$ pairs. We study the complementary question of equality, namely, identifying nontrivial families of copulas $C$ satisfying ${\tau }_C={\rho }_C$. We prove that this equality holds for every factorizable copula of the form $C_{e,\alpha }\ast C_{\beta ,e}$, where $\alpha$ and $\beta$ 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.