Title: Semiparametric Bayesian estimation in copula models

Abstract: Approximate Bayesian computation (ABC) is a recent class of algorithms which allow for managing complex models where the likelihood function may be considered intractable. Complex models are usually characterized by a dependence structure difficult to model with the usual tools. Copula models have been introduced as a probabilistic way to describe general multivariate distributions by considering the marginal distributions and a copula function which captures the dependence structure among the components of the vector. While it is often straightforward to produce reliable estimates for the marginals, estimating the dependence structure is more complicated, in particular in high dimensional problems, nevertheless it is crucial. Major areas of application include econometrics, engineering, biomedical science, signal processing and finance. We consider the general problem of estimating some specific quantities of interest of a generic copula (such as, for example, tail dependence indices or rank correlation coefficients) by adopting an approximate Bayesian approach based on computing the empirical likelihood as an approximation of the likelihood function for the quantity of interest. This approach is general, in the sense that it could be adapted to both parametric and nonparametric modeling of the marginal distributions and on a parametric or semiparametric estimation of the copula function. We will show how the Bayesian procedure based on ABC shows better properties that the classical inferential solutions available in the literature and apply the method in both simulated and real examples.