Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom

In mathematical finance and other applications of statistics, the computation of expectations is often taken over a multi-dimensional probability distribution where there is no clear multivariate distribution. Copula theory has become increasingly popular as a means of gluing marginals together to circumvent this difficulty. There is then the issue of reconciling the distributions implied by various choices of copula and marginal with candidates for the canonical multivariate distribution when such candidates become available. This article looks at the copulae and candidate multivariate distributions for a general multivariate Student’s T distribution when the marginals do not necessarily have the same degrees of freedom. We discuss the grouped T copula proposed recently by Demarta and McNeil, and Daul et al and other options, including one based on our own generalization of recent work by Jones, and a further proposal of our own. We compare these with the meta-elliptical distributions proposed as the canonical multivariate distribution by Fang et al. We argue that the natural appearance of independence in the zero-dependence case should take priority over preserving the elliptical structure commonplace in multivariate distribution theory. We are able to give several detailed and explicit representations for the bivariate case. For the bivariate case where one distribution is Normal we argue that there is indeed a canonical bivariate Student-Normal distribution with a naturally associated copula that arises simultaneously from several of the copula methods, and an elegant tractable density is available. For the StudentStudent case there appears to be some genuine choice as to the canonical distribution, though the requirement of independence in the zero-correlation case appears to constrain us to just one choice. We also briefly discuss the inclusion of correlation data relevant to calibration.