Maximal asymmetry of bivariate copulas and consequences to measures of dependence

Abstract

Abstract In this article, we focus on copulas underlying maximal non-exchangeable pairs ( X , Y ) \left(X,Y) of continuous random variables X , Y X,Y either in the sense of the uniform metric d ∞ {d}_{\infty } 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 {\zeta }_{1} and ξ \xi . Considering maximal d ∞ {d}_{\infty } -asymmetry we obtain ζ 1 ∈ 5 6 , 1 {\zeta }_{1}\in \left[\frac{5}{6},1\right] and ξ ∈ 2 3 , 1 \xi \in \left[\frac{2}{3},1\right] , and in the case of maximal D 1 {D}_{1} -asymmetry we obtain ζ 1 ∈ 3 4 , 1 {\zeta }_{1}\in \left[\frac{3}{4},1\right] and ξ ∈ 1 2 , 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 D 1 {D}_{1} -asymmetry and derive some surprising properties for maximal D p {D}_{p} -asymmetric copulas.