Maximal asymmetry of bivariate copulas and consequences to measures of dependence

In this article, we focus on copulas underlying maximal non-exchangeable pairs (X,Y)\left(X,Y) of continuous random variables X,YX,Y either in the sense of the uniform metric d∞{d}_{\infty } or the conditioning-based metrics Dp{D}_{p}, and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ1{\zeta }_{1} and ξ\xi . Considering maximal d∞{d}_{\infty }-asymmetry we obtain ζ1∈56,1{\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ∈23,1\xi \in \left[\frac{2}{3},1\right], and in the case of maximal D1{D}_{1}-asymmetry we obtain ζ1∈34,1{\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ∈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 maximal D1{D}_{1}-asymmetry and derive some surprising properties for maximal Dp{D}_{p}-asymmetric copulas.