Objective priors for the number of degrees of freedom of a multivariate t distribution and the t-copula

Abstract

We propose an objective Bayesian approach to estimate the number of degrees of freedom for the multivariate \(t\) distribution and for the \(t\)-copula, when the parameter is considered discrete. Inference on this parameter has been problematic, as the scarce literature for the multivariate \(t\) shows and, more important, the absence of any method for the \(t\)-copula. We employ an objective criterion based on loss functions which allows to overcome the issue of defining objective probabilities directly. The truncation derives from the property of both the multivariate \(t\) and the \(t\)-copula to convergence to normality for a sufficient large number of degrees of freedom. The performance of the priors is tested on simulated scenarios and on real data: daily logarithmic returns of IBM and of the Center for Research in Security Prices Database.