Generalized factorizability of certain implicit dependence copulas

Abstract

The copula CX,Y of random variables X and Y that are uniformly distributed on [0,1] is called an implicit dependence copula if f(X)=g(Y) almost surely for some Borel functions f and g on [0,1]. Via a generalized Markov product, we give a one-to-one correspondence between the implicit dependence copulas and the parametric classes of subcopulas on a corresponding domain for the cases that f=g=Λθ, the tent function whose top is at (θ,1). We also show in the case f=g=α, a simple measure-preserving function, that implicit dependence copulas are generalized factorizable.