On differentiability and mass distributions of typical bivariate copulas

Abstract

Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via $(d-1)$-increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas $C$ whose first partial derivative $\partial_1C(x,y)$ is pathological in the sense that for almost every $x \in (0,1)$ it does not exist on a dense subset of $y \in (0,1)$, and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVC) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically $\partial_1C(x,y)$ exists in full $(0,1)^2$. Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures $\vartheta$ on $[0,1]$, show that regularity of $\vartheta$ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.