Concordance, symmetrization and non-exchangeability for bivariate copulas

We study the relationship between measures of non-exchangeability ${\mu }_p$ ($p\in [1,+\mathrm{\infty }]$), in the sense of Durante et al. (2010), and classical dependence functionals for bivariate copulas. We show that the symmetrization $C\mapsto (C+C^t)\mathrm{/}2$ 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.