Distribution functions of copulas: a class of bivariate probability integral transforms

We discuss a two-dimensional analog of the probability integral transform for bivariate distribution functions H1 and H2, i.e., the distribution function of the random variable H1(X,Y) given that the joint distribution function of the random variables X and Y is H2. We study the case when H1 and H2 have the same continuous marginal distributions, showing that the distribution function of H1(X,Y) depends only on the copulas C1 and C2 associated with H1 and H2. We examine various properties of these "distribution functions of copulas", and illustrate applications including dependence orderings and measures of association.