We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Further, the continuity of the imposed D-P tree prior leads to a more favorable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007-08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviors. a novel multi-partition Dirichlet-based Pólya tree (D-P tree) prior on the copula. Our D-P tree prior relaxes the binary partition constraints on earlier Pólya-tree-like priors but still preserves the favorable properties of the Pólya tree, including conjugacy and absolute continuity. Based on such a D-P tree prior, we provide a nonparametric Bayesian approach for copula estimation. Its consistency is validated through theoretical analysis.The D-P tree prior overcomes the severe bias problem of previously proposed Pólya-tree-like priors, and the inconsistency issue of family-based nonparametric Bayesian approaches such as the Gaussian copula mixture (Dortet-Bernadet 2005) under model misspecification. Further, compared with classic nonparametric frequentist methods, including the empirical copula estimation and the kernel method, the D-P tree shows a more favorable smoothing effect, especially based on small sets of observations. We illustrate our new method by focusing on copula structure prediction between the S&P 500 daily index and the IBM daily stock prices during the 2007-08 financial crisis. We find that D-P tree-based methods are rather robust and adaptive to irregular market behavior, especially in comparison with commonly-adopted parametric models and the empirical method.Earlier parametric or semi-parametric methods often model copula functions within certain parametric copula families and estimate the parameters by maximum likelihood (ML). For marginals, either parametric or nonparametric estimations are usually adopted (Joe 1997;Jaworski et al. 2010;Chen and Huang 2007;Oakes 1982 Oakes 1986Genest et al. 1995). However, these parametric or semiparametric methods suffer from the risk of severe bias when the model is misspecified, thus lack the flexibility to provide accurate estimation for more complex and subtle copula structures. In addition, copula itself is strictly-increasing-transform invariant (Schweizer and Wolff 1981). Thereby, under no further parametric assumptions, the rank statistics of data would preserve sufficient information required for the estimation. In light of these features...
