Multivariate Student distribution

In statistics, a multivariate Student distribution is a multivariate generalization of the Student's t-distribution. One common method of construction, for the case of $$p$$ dimensions, is based on the observation that if $${\mathbf y}$$ and $$u$$ are independent and distributed as $${\mathcal N}({\mathbf 0},{\mathbf\Sigma})$$ and $$\chi^2_n$$ respectively, $$\mathbf{\Sigma}\,$$ is a p x p matrix, and $${\mathbf y}\sqrt{n/u}={\mathbf x}-{\mathbf\mu}$$, then $${\mathbf x}$$  has the density



\frac{\Gamma\left[(n+p)/2\right]}{\Gamma(n/2)n^{p/2}\pi^{p/2}\left|{\mathbf\Sigma}\right|^{1/2}\left[1+\frac{1}{n}({\mathbf x}-{\mathbf\mu})^T{\mathbf\Sigma}^{-1}({\mathbf x}-{\mathbf\mu})\right]^{(n+p)/2}}$$

and is said to be distributed as a Multivariate t-distribution with parameters $${\mathbf\Sigma},{\mathbf\mu},n$$.

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 ($$p=1$$), with $$t=x$$, we have the probability density function
 * $$f(t) = \frac{\Gamma[(n+1)/2]}{\sqrt{n\pi\,}\,\Gamma[n/2]} (1+t^2/n)^{-(n+1)/2}$$

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 $$p$$ variables $$t_i$$ that replaces $$t^2$$ by a quadratic function of all the $$t_i$$. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom $$n$$. With $$ A = \Sigma^{-1}$$, one has a simple choice of multivariate density function
 * $$f(t_i) = \frac{\Gamma((n+p)/2)\left|A\right|^{1/2}}{\sqrt{n^p\pi^p\,}\,\Gamma(n/2)} (1+\sum_{i,j=1}^{p,p} A_{ij} t_i t_j/n)^{-(n+p)/2}$$

which is the standard, but non the only, choice.

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


 * $$f(t_i) = \frac{\left|A\right|^{1/2}}{2\pi} (1+\sum_{i,j=1}^{2,2} A_{ij} t_i t_j/n)^{-(n+2)/2}$$

and if $$A$$ is the identity matrix we have


 * $$f(t_i) = \frac{1}{2\pi} (1+(t_1^2 + t_2^2)/n)^{-(n+2)/2}.$$

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 $$ \Sigma$$ is diagonal the standard representation can be shown to have zero correlation but there is still dependency between the marginal distributions. There are differing views on this issue, which is under discussion in the research literature as of early 2007.

Further theory
Many such distributions may be constructed by considering the quotients of normal random variables with the square root of a sample from a chi-squared distribution. These are surveyed in the references and links below.

Copulas based on the multivariate Student
The use of such distributions is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student t copula.