Multivariate Student Distribution - Derivation

Derivation

There are in fact many candidates for the multivariate generalization of Student's t-distribution. An extensive survey of the field has been given by Kotz and Nadarajah (2004). The essential issue is to define a probability density function of several variables that is the appropriate generalization of the formula for the univariate case. In one dimension, with and, we have the probability density function

and one approach is to write down a corresponding function of several variables. This is the basic idea of elliptical distribution theory, where one writes down a corresponding function of variables that replaces by a quadratic function of all the . It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom . With, one has a simple choice of multivariate density function

which is the standard but not the only choice.

An important special case is the standard bivariate t-distribution, p = 2:

and, if is the identity matrix, the density is

The difficulty with the standard representation is revealed by this formula, which does not factorize into the product of the marginal one-dimensional distributions. When is diagonal the standard representation can be shown to have zero correlation but the marginal distributions are not statistically independent. There are differing views on this issue, which is under discussion in the research literature as of early 2007.