Concordance, symmetrization and non-exchangeability for bivariate copulas

TL;DR

This paper explores the relationship between non-exchangeability measures and classical dependence functionals in bivariate copulas, revealing how symmetrization affects these measures and deriving bounds and explicit formulas for dependence measures.

Contribution

It establishes new theoretical links between non-exchangeability and dependence functionals, including bounds and explicit formulas for maximally non-exchangeable copulas.

Findings

Symmetrization preserves Spearman's ρ but nullifies non-exchangeability measures.

Blomqvist's β provides no information on non-exchangeability.

A sharp lower bound relates dependence measure σ to non-exchangeability μ₁.

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.