Joining copulas of extreme implicit dependence copulas

Abstract

Copulas of uniform-$(0,1)$ random variables U and V satisfying $\alpha \left(U\right)=\beta \left(V\right)$ almost surely for some measure-preserving transformations α and β are called implicit dependence copulas. They were recently shown to coincide with the generalized Markov products of $C_{e,\alpha }$ and $C_{\beta ,e}$ with respect to a class of joining copulas ${\left(A_t\right)}_{t\in [0,1]}$. If $C_{e,\alpha }$ and $C_{\beta ,e}$ are not two-sided invertible, then most implicit dependence copulas, especially when $A_t\equiv \Pi$, are not extreme points in the class of copulas. For a given pair of left and right invertible copulas $C_{e,\alpha }$ and $C_{\beta ,e}$, we characterize extreme implicit dependence copulas in terms of the extremality of the joining copulas in the class of subcopulas on a domain involving the invertible copulas. This result is then extended to the multivariate case.