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

## Definition

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

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

and is said to be distributed as a multivariate t-distribution with parameters ${\displaystyle {\boldsymbol {\Sigma }},{\boldsymbol {\mu }},\nu }$. Note that ${\displaystyle \mathbf {\Sigma } }$ is not the covariance matrix since the covariance is given by ${\displaystyle \nu /(\nu -2)\mathbf {\Sigma } }$ (for ${\displaystyle \nu >2}$).

In the special case ${\displaystyle \nu =1}$, 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 (${\displaystyle p=1}$), with ${\displaystyle t=x-\mu }$ and ${\displaystyle \Sigma =1}$, we have the probability density function

${\displaystyle f(t)={\frac {\Gamma [(\nu +1)/2]}{{\sqrt {\nu \pi \,}}\,\Gamma [\nu /2]}}(1+t^{2}/\nu )^{-(\nu +1)/2}}$

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 ${\displaystyle p}$ variables ${\displaystyle t_{i}}$ that replaces ${\displaystyle t^{2}}$ by a quadratic function of all the ${\displaystyle t_{i}}$. It is clear that this only makes sense when all the marginal distributions have the same degrees of freedom ${\displaystyle \nu }$. With ${\displaystyle \mathbf {A} ={\boldsymbol {\Sigma }}^{-1}}$, one has a simple choice of multivariate density function

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

which is the standard but not the only choice.

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

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

Note that ${\displaystyle {\frac {\Gamma \left({\frac {\nu +2}{2}}\right)}{\pi \ \nu \Gamma \left({\frac {\nu }{2}}\right)}}={\frac {1}{2\pi }}}$.

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

${\displaystyle f(t_{1},t_{2})={\frac {1}{2\pi }}\left(1+(t_{1}^{2}+t_{2}^{2})/\nu \right)^{-(\nu +2)/2}.}$

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 ${\displaystyle \Sigma }$ 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 ${\displaystyle \mathbf {x} }$ is a real vector):

${\displaystyle F(\mathbf {x} )=\mathbb {P} (\mathbf {X} \leq \mathbf {x} ),\quad {\textrm {where}}\;\;\mathbf {X} \sim t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}).}$

There is no simple formula for ${\displaystyle F(\mathbf {x} )}$, but it can be approximated numerically via Monte Carlo integration.[1][2]

## 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.