Inference on Archimedean copulas using mixtures of Polya trees

Assume that X = (X-1,...,X-d), d >= 2 is a random vector having joint cumulative distribution function H with continuous marginal cumulative distribution functions F-1,.., F-d respectively. Sklar's decomposition yields a unique copula C such that H(chi(1),..., chi(d)) = C(F-1 (chi(1)), ... , F-d(chi(d))) for all (chi(1), ... , chi(d)) is an element of R-d. Here F-1, ... , F-d and C are the unknown parameters, the one of interest being the copula C. We assume C to belong to the Archimedean family, that is C = C-psi, for some Archimedean generator psi. We exploit the well known fact that such a generator is in one-to-one correspondence with the distribution function of a nonnegative random variable R with no atom at zero. In order to adopt a Bayesian approach for inference, a prior on the Archimedean family may be selected via a prior on the cumulative distribution function F of R. A mixture of Polya trees is proposed for F, making the model very flexible, yet still manageable. The induced prior is concentrated on the space of absolutely continuous d-dimensional Archimedean copulas and explicit forms for the generator and its derivatives are available. To the best of our knowledge, others in the literature have not yet considered such an approach. An extensive simulation study is carried out to compare our estimator with a popular frequentist nonparametric estimator. The results clearly indicate that if intensive computing is available, our estimator is worth considering, especially for small samples. (C) 2014 Elsevier B.V. All rights reserved.