Basics of Multivariate Student’s t distribution


I am a fan of the Student’s t distribution – it is almost as easy to handle as the normal distribution but has an additional flexibility with respect to the heaviness of the tails. In fact, the normal distribution is a special case and so is the Cauchy distribution.

The density of a (non-degenerate) multivariate normal distribution with zero mean is given by

\frac{1}{\sqrt{(2\pi)^n |\Sigma|}} \exp \Big( -\frac{1}{2} x^\top \Sigma^{-1} x \Big),

where \Sigma is a (positive-semidefinite) n\times n covariance matrix.

For comparison, the density of the multivariate student t distribution with zero mean is

\frac{\Gamma[(\nu + n)/2]}{\Gamma(\nu/2) \sqrt{v^{n} \pi^{n} |\Sigma|}} \Big[ 1 + \frac{1}{\nu} x^\top \Sigma^{-1} x \Big]^{-(\nu + n)/2}.

Compare this with the univariate version of the t density:

\frac{\Gamma{(\nu +1)/2}}{\Gamma(\nu/2)\sqrt{\nu\pi}} \Big[ 1 + \frac{x^2}{2} \Big]^{-(\nu+1)/2}.

In both cases, \nu>2 is the degrees of freedom parameter.

A standard way to construct a random variable Z with t-distribution is to use a normal random variable X and an independent chi-square random variable Y and set

Z = \frac{X}{\sqrt{Y/\nu}}.

This formula works for both the univariate and the multivariate case and can be used to draw samples from the t distribution using independent samples of X and Y. Also, we get

\text{Cov} Z = \frac{\nu}{\nu-2} \text{Cov} X.

This is because the univariate \chi^2 distribution with \nu degrees of freedom is really a \text{Gamma}(\frac{\nu}{2}, \frac{1}{2}) distribution. This can be seen by comparing a \chi^2 density with the \text{Gamma}(\alpha, \beta) density

p_{\alpha,\beta}(x) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x} 1_{[0, \infty)}(x).

The parameters \alpha and \beta are both positive. Moreover, 1/Y is inverse-gamma distributed and thus \mathbb E \frac{1}{Y} = \frac{1}{\nu -2}.