# Multivariate t-distribution

In statistics, the multivariate t-distribution (or multivariate Student distribution) is a multivariate probability distribution. It is a generalization to random vectors of the Student's t-distribution, which is a distribution applicable to univariate random variables. While the case of a random matrix could be treated within this structure, the matrix t-distribution is distinct and makes particular use of the matrix structure.

Notation $t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})$ ${\boldsymbol {\mu }}=[\mu _{1},\dots ,\mu _{p}]^{T}$ location (real $p\times 1$ vector)${\boldsymbol {\Sigma }}$ shape matrix (positive-definite real $p\times p$ matrix) $\nu$ is the degrees of freedom $\mathbf {x} \in \mathbb {R} ^{p}\!$ ${\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{\rm {T}}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}$ No analytic expression, but see text for approximations ${\boldsymbol {\mu }}$ if $\nu >1$ ; else undefined ${\boldsymbol {\mu }}$ ${\boldsymbol {\mu }}$ ${\frac {\nu }{\nu -2}}{\boldsymbol {\Sigma }}$ if $\nu >2$ ; else undefined 0

## Definition

One common method of construction of a multivariate t-distribution, for the case of $p$ dimensions, is based on the observation that if $\mathbf {y}$ and $u$ are independent and distributed as ${\mathcal {N}}({\mathbf {0} },{\boldsymbol {\Sigma }})$ and $\chi _{\nu }^{2}$ (i.e. multivariate normal and chi-squared distributions) respectively, the matrix $\mathbf {\Sigma } \,$ is a p × p matrix, and ${\mathbf {y} }/{\sqrt {u/\nu }}={\mathbf {x} }-{\boldsymbol {\mu }}$ , then ${\mathbf {x} }$ has the density

${\frac {\Gamma \left[(\nu +p)/2\right]}{\Gamma (\nu /2)\nu ^{p/2}\pi ^{p/2}\left|{\boldsymbol {\Sigma }}\right|^{1/2}}}\left[1+{\frac {1}{\nu }}({\mathbf {x} }-{\boldsymbol {\mu }})^{T}{\boldsymbol {\Sigma }}^{-1}({\mathbf {x} }-{\boldsymbol {\mu }})\right]^{-(\nu +p)/2}$ and is said to be distributed as a multivariate t-distribution with parameters ${\boldsymbol {\Sigma }},{\boldsymbol {\mu }},\nu$ . Note that $\mathbf {\Sigma }$ is not the covariance matrix since the covariance is given by $\nu /(\nu -2)\mathbf {\Sigma }$ (for $\nu >2$ ).

In the special case $\nu =1$ , the distribution is a multivariate Cauchy distribution.

## 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 ($p=1$ ), with $t=x-\mu$ and $\Sigma =1$ , we have the probability density function

$f(t)={\frac {\Gamma [(\nu +1)/2]}{{\sqrt {\nu \pi \,}}\,\Gamma [\nu /2]}}(1+t^{2}/\nu )^{-(\nu +1)/2}$ 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 $\nu$ . With $\mathbf {A} ={\boldsymbol {\Sigma }}^{-1}$ , one has a simple choice of multivariate density function

$f(\mathbf {t} )={\frac {\Gamma ((\nu +p)/2)\left|\mathbf {A} \right|^{1/2}}{{\sqrt {\nu ^{p}\pi ^{p}\,}}\,\Gamma (\nu /2)}}\left(1+\sum _{i,j=1}^{p,p}A_{ij}t_{i}t_{j}/\nu \right)^{-(\nu +p)/2}$ which is the standard but not the only choice.

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

$f(t_{1},t_{2})={\frac {\left|\mathbf {A} \right|^{1/2}}{2\pi }}\left(1+\sum _{i,j=1}^{2,2}A_{ij}t_{i}t_{j}/\nu \right)^{-(\nu +2)/2}$ Note that ${\frac {\Gamma \left({\frac {\nu +2}{2}}\right)}{\pi \ \nu \Gamma \left({\frac {\nu }{2}}\right)}}={\frac {1}{2\pi }}$ .

Now, if $\mathbf {A}$ is the identity matrix, the density is

$f(t_{1},t_{2})={\frac {1}{2\pi }}\left(1+(t_{1}^{2}+t_{2}^{2})/\nu \right)^{-(\nu +2)/2}.$ 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 the marginal distributions do not agree with statistical independence.

### Cumulative distribution function

The definition of the cumulative distribution function (cdf) in one dimension can be extended to multiple dimensions by defining the following probability (here $\mathbf {x}$ is a real vector):

$F(\mathbf {x} )=\mathbb {P} (\mathbf {X} \leq \mathbf {x} ),\quad {\textrm {where}}\;\;\mathbf {X} \sim t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}).$ There is no simple formula for $F(\mathbf {x} )$ , but it can be approximated numerically via Monte Carlo integration.

## 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 t

The use of such distributions is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student's t copula.

In univariate statistics, the Student's t-test makes use of Student's t-distribution. Hotelling's T-squared distribution is a distribution that arises in multivariate statistics. The matrix t-distribution is a distribution for random variables arranged in a matrix structure.

• Multivariate normal distribution, which is a special case of the multivariate Student's t-distribution when $\nu \uparrow \infty$ .