Inference on Archimedean copulas using mixtures of Pólya trees

Assume that X=(X1,…,Xd)X=(X1,…,Xd), d⩾2d⩾2 is a random vector having joint cumulative distribution function HH with continuous marginal cumulative distribution functions F1,…,FdF1,…,Fd respectively. Sklar’s decomposition yields a unique copula CC such that H(x1,…,xd)=C(F1(x1),…,Fd(xd))H(x1,…,xd)=C(F1(x1),…,Fd(xd)) for all (x1,…,xd)∈Rd(x1,…,xd)∈Rd. Here F1,…,FdF1,…,Fd and CC are the unknown parameters, the one of interest being the copula CC. We assume CC to belong to the Archimedean family, that is C=CψC=Cψ, for some Archimedean generator ψψ. 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 RR 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 FF of RR. A mixture of Pólya trees is proposed for FF, making the model very flexible, yet still manageable. The induced prior is concentrated on the space of absolutely continuous dd-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.