Concordance, symmetrization and non-exchangeability for bivariate copulas

Abstract

We study the relationship between measures of non-exchangeability ${\mu }_p$ ($p\in [1,+\infty ]$), in the sense of Durante et al. (2010), and classical dependence functionals for bivariate copulas. We show that the symmetrization $C\mapsto (C+C^t)/2$ preserves Spearman's $\rho$ while annihilating ${\mu }_p$, and that Blomqvist's $\beta$ carries no information about the degree of non-exchangeability. We also establish the sharp lower bound $\sigma (C)\ge 6{\mu }_1(C)$, where $\sigma$ is the Schweizer-Wolff dependence measure, showing that asymmetry implies dependence. Closed-form expressions for $\tau$, $\rho$, and the tail-dependence coefficients of the maximally non-exchangeable family $\{M_{\theta }\}$ are derived as illustrations.

arXiv commentThe authors have chosen to withdraw this manuscript because a potential flaw was identified in the main result. Further investigation and refinement are required to ensure the validity of the findings. The authors intend to update and resubmit the work once these issues are fully resolved