Extremality of factorizable copulas and implicit dependence copulas

Abstract

Copulas of continuous random variables satisfying f(X)=g(Y) a.s. for some Borel functions f and g, where f(X) has continuous distribution, are called implicit dependence copulas, of which a special subclass is the class of factorizable copulas, i.e., copulas that can be written as the Markov product of a left invertible copula and a right invertible copula in that order. Implicit dependence copulas were recently shown to coincide with generalized Markov products of left and right invertible copulas, where the joining copulas are arbitrary. In this paper, we obtain characterizations of extreme factorizable copulas and prove that implicit dependence copulas whose joining copula is one of the Fréchet-Hoeffding bounds are extreme. This condition, however, is not necessary, which will be shown via some examples of extreme implicit dependence copulas.