Definition: Multivariate t-distribution

Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate continuous distributions ▷ Multivariate t-distribution ▷ Definition

Definition: Let $X$ be an $n \times 1$ random vector. Then, $X$ is said to follow a multivariate $t$-distribution with mean $\mu$, scale matrix $\Sigma$ and degrees of freedom $\nu$

\[\label{eq:mvt} X \sim t(\mu, \Sigma, \nu) \; ,\]

if and only if its probability density function is given by

\[\label{eq:mvt-pdf} t(x; \mu, \Sigma, \nu) = \sqrt{\frac{1}{(\nu \pi)^{n} |\Sigma|}} \, \frac{\Gamma([\nu+n]/2)}{\Gamma(\nu/2)} \, \left[ 1 + \frac{1}{\nu} (x-\mu)^\mathrm{T} \Sigma^{-1} (x-\mu) \right]^{-(\nu+n)/2}\]

where $\mu$ is an $n \times 1$ real vector, $\Sigma$ is an $n \times n$ positive definite matrix and $\nu > 0$.

 

Sources:

• Koch KR (2007): "Multivariate t-Distribution"; in: Introduction to Bayesian Statistics, ch. 2.5.2, pp. 53-55; URL: https://www.springer.com/gp/book/9783540727231; DOI: 10.1007/978-3-540-72726-2.

Metadata: ID: D148 | shortcut: mvt | author: JoramSoch | date: 2020-04-21, 08:16.