A copula can fully characterize the dependence of multiple variables. A typical issue in statistics is dealing with high dimension in copula estimation. Particularly, with vine copula construction it turns out the needing of model selection for each pair copula. Thus, the higher is the dimension of the joint distribution, the higher is the computational cost. We propose a Bayesian nonparametric method to estimate the pair copulas in the vine, extending the approach presented by Wu et. al (2015). For each pair copula we use an infinite mixture of Gaussian copula densities to define a nonparametric copula for modelling any dependence structure between the marginals. Therefore, we assume a Dirichlet process prior on each pair copula. Our approach has two main advantages compared to the traditional methods: on the one hand it is extremely flexible, due to the vine structure, and on the other hand it overcomes the need of specify the families of each pair copula.
In order to allow for maximum flexibility in a d-dimensional framework, with d>2, we first choose a vine copula construction for modelling the joint distribution of the marginals. With a Bayesian nonparametric method we avoid copula model selection, which might be problematic and controversial in several applications, getting a substantial reduction of the computational cost.
For the copula modelling, we propose to extend the method proposed by Wu et. al (2013) to the vine copula case. We would like to model each pair as an infinite mixture of Gaussian copula function, including pair copulas having as margins the conditional distribution functions. Thus, we extend the methodology proposed by Wu et. al to the vine copula framework.
We assume a non-informative Dirchlet process prior on each pair copula: Gaussian copulas parameters for each pair are estimated via Gibbs sampling. Particularly, as in Dalla Valle et. al (2017), the sampling strategy follows the slice sampler of Walker (2007) and Kalli et al. (2011). The posterior pair-copula are estimated via Kernel density estimation.