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) (X,Y) of continuous random variables X, Y X,Y either in the sense of the uniform metric d ∞ or the conditioning-based metrics D p {D}_{p}, and analyze their possible extent of dependence quantified by the recently introduced dependence measures ζ 1 and ζ \xi. Considering maximal d ∞-Asymmetry we obtain ζ 1 5 6, 1] and ζ 2 3,1], and in the case of maximal D 1 {D}_{1}-Asymmetry we obtain ζ 1 3 4, 1 and ζ 1 2, 1, 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 D 1 {D}_{1}-Asymmetry and derive some surprising properties for maximal Dp-Asymmetric copulas.