Multivariate t-distribution

Multivariate t

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

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.

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.

Related concepts

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.

See also

• Multivariate normal distribution, which is a special case of the multivariate Student's t-distribution when $\nu \uparrow \infty$.
• Chi distribution, the pdf of the scaling factor in the construction the Student's t-distribution and also the 2-norm (or Euclidean norm) of a multivariate normally distributed vector (centered at zero).
• Mahalanobis distance

Literature

• Kotz, Samuel; Nadarajah, Saralees (2004). Multivariate t Distributions and Their Applications. Cambridge University Press. ISBN 978-0521826549.
• Cherubini, Umberto; Luciano, Elisa; Vecchiato, Walter (2004). Copula methods in finance. John Wiley & Sons. ISBN 978-0470863442.

External links

• Copula Methods vs Canonical Multivariate Distributions: the multivariate Student T distribution with general degrees of freedom
• Multivariate Student's t-distribution