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In this article, we focus on copulas underlying maximal non-exchangeable pairs(X,Y)\left(X,Y)of continuous random variablesX,YX,Yeither in the sense of the uniform metricd∞{d}_{\infty }or the conditioning-based metricsDp{D}_{p}, and analyze their possible extent of dependence quantified by the recently introduced dependence measuresζ1{\zeta }_{1}andξ\xi. Considering maximald∞{d}_{\infty }-asymmetry we obtainζ1∈56,1{\zeta }_{1}\in \left[\frac{5}{6},1\right]andξ∈23,1\xi \in \left[\frac{2}{3},1\right], and in the case of maximalD1{D}_{1}-asymmetry we obtainζ1∈34,1{\zeta }_{1}\in \left[\frac{3}{4},1\right]andξ∈12,1\xi \in \left(\frac{1}{2},1\right], implying that maximal asymmetry implies a very high degree of dependence in both cases. Furthermore, we study various topological properties of the family of copulas with maximalD1{D}_{1}-asymmetry and derive some surprising properties for maximalDp{D}_{p}-asymmetric copulas.

Maximal asymmetry of bivariate copulas and consequences to measures of dependence