Bayesian nonparametric estimation of a copula

Abstract

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.

Bayesian Nonparametric Conditional Copula Estimation of Twin Data

“…In this section, we review some preliminary notions about conditional copulas and illustrate the Bayesian non-parametric copula density estimation that was introduced in Wu et al (2015). In what follows, we focus on the bivariate case for simplicity; however, the arguments can be easily extended to more than two dimensions.…”

“…where Φ is the univariate standard normal distribution function. The Gaussian copula density is Wu et al (2015) proposed to use an infinite mixture of Gaussian copulas for the estimation of a copula density, as follows:…”

“…The data object of study requires a model which can take into account the effect of a covariate. We build on the model that was introduced by Wu et al (2015) and illustrated in the previous section. The idea is to replace the Gaussian copula with a conditional version where the correlation is a function of the covariate, i.e.…”

“…In the numerical experiments, we consider a base measure G 0 which is a multivariate normal distribution with zero-mean vector and variance σ 2 I d , where I d is the d-dimensional identity matrix and σ 2 > 0. As in Wu et al (2015), our model can be described as an infinite mixture of normal distributions,…”

“…In this paper, we adopt a Bayesian non-parametric approach which enables us to overcome the issue of the choice of copula and we adopt a conditional copula approach to model the effect of a covariate on the dependence between variables. Our methodology builds on Wu et al (2015), who proposed a Bayesian non-parametric procedure to estimate any unconditional copula density function. They combined the well-known Gaussian copula density with the modelling flexibility of the Bayesian non-parametric approach, proposing to use an infinite mixture of Gaussian copulas.…”