A copula can characterize the complete dependence of multi-variables separately from the univariate marginals. The purpose of this paper is to provide a Bayesian nonparametric methodology to estimate a copula. We show that any bivariate copula density can be approximated by an infinite mixture of the Gaussian copula densities that are the dependence structures of the pairs with standard normal marginals. A slice sampling idea is introduced for this infinite structure that can estimate the number of occupied clusters in a sampler. The estimation procedure is proposed by the Gibbs sampling algorithm. Simulation and real data application illustrate the rational of the proposed approach.

The paper presents a general Bayesian nonparametric approach for estimating a high dimensional copula. We first introduce the skew-normal copula, which we then extend to an infinite mixture model. The skew-normal copula fixes some limitations in the Gaussian copula. An MCMC algorithm is developed to draw samples from the correct posterior distribution and the model is investigated using both simulated and real applications. Consistency of the Bayesian nonparametric model is established.

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