Multivariate t-Distribution
Definition
Given a location vector $\mathbf{\mu} \in \mathbb{R}^{p}$ and a scale matrix $\Sigma \in \mathbb{R}^{p \times p}$ that is positive definite, the multivariate distribution $t_{p} \left(\nu; \mu , \Sigma \right)$ with the following probability density function is referred to as the Multivariate t-distribution.
$$ f (\textbf{x}) = {{ \Gamma \left[ (\nu + p) / 2 \right] } \over { \Gamma ( \nu / 2) \sqrt{ \nu^{p} \pi^{p} \det \Sigma } }} \left[ 1 + {{ 1 } \over { \nu }} \left( \textbf{x} - \mathbf{\mu} \right)^{T} \Sigma^{-1} \left( \textbf{x} - \mathbf{\mu} \right) \right] \qquad , \textbf{x} \in \mathbb{R}^{p} $$
Description
- When $p = 1$, then $\mu \in \mathbb{R}^{1}$, and $\Sigma \in \mathbb{R}^{1 \times 1}$, the above probability density function precisely becomes the probability density function of a univariate t-distribution with degrees of freedom $\nu$.
- Just as when $\nu = 1$ the t-distribution turns into a Cauchy distribution, the multivariate t-distribution likewise becomes a Multivariate Cauchy distribution.