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

Abstract

An objective Bayesian approach to estimate the number of degrees of freedom (ν) for the multivariate t distribution and for the t-copula, when the parameter is considered discrete, is proposed. Inference on this parameter has been problematic for the multivariate t and, for the absence of any method, for the t-copula. An objective criterion based on loss functions which allows to overcome the issue of defining objective probabilities directly is employed. The support of the prior for ν is truncated, which derives from the property of both the multivariate t and the t-copula of convergence to normality for a sufficiently large number of degrees of freedom. The performance of the priors is tested on simulated scenarios 1 and on real data: daily logarithmic returns of IBM and of the Center for Research in Security Prices Database.