Copulas, marginals, and joint distributions

Abstract

Any pair of univariate marginal distributions can be combined with any copula to yield a bivariate distribution with the given marginals.This being the case, it is tempting to conclude that the dependence properties of the distribution can be determined by examination of the copula alone.Unfortunately, this idea is seriously flawed: (i) Copulas exist which yield the Frechet upper bound for some marginal pairs and the Frechet lower bound for other marginal pairs, (ii) There is no nonconstant measure of dependence which depends only on the copula, (iii) Weakly convergent sequences of bivariate distributions with continuous marginals exist for which the unique corresponding copulas do not converge.Related issues are considered.

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