# Multivariate t-distribution

Notation ${\displaystyle t_{\nu }({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}$ ${\displaystyle {\boldsymbol {\mu }}=[\mu _{1},\dots ,\mu _{p}]^{T}}$ location (real ${\displaystyle p\times 1}$ vector)${\displaystyle {\boldsymbol {\Sigma }}}$ scale 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

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 ${\displaystyle p}$ dimensions, is based on the observation that if ${\displaystyle \mathbf {y} }$ and ${\displaystyle u}$ are independent and distributed as ${\displaystyle N({\mathbf {0} },{\boldsymbol {\Sigma }})}$ 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 ${\displaystyle {\boldsymbol {\mu }}}$ is a constant vector then the random variable ${\textstyle {\mathbf {x} }={\mathbf {y} }/{\sqrt {u/\nu }}+{\boldsymbol {\mu }}}$ has the density[1]

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

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 ${\displaystyle \nu >2}$).

The constructive definition of a multivariate t-distribution simultaneously serves as a sampling algorithm:

1. Generate ${\displaystyle u\sim \chi _{\nu }^{2}}$ and ${\displaystyle \mathbf {y} \sim N(\mathbf {0} ,{\boldsymbol {\Sigma }})}$, independently.
2. Compute ${\displaystyle \mathbf {x} \gets {\sqrt {\nu /u}}\mathbf {y} +{\boldsymbol {\mu }}}$.

This formulation gives rise to the hierarchical representation of a multivariate t-distribution as a scale-mixture of normals: ${\displaystyle u\sim \mathrm {Ga} (\nu /2,\nu /2)}$ where ${\displaystyle \mathrm {Ga} (a,b)}$ indicates a gamma distribution with density proportional to ${\displaystyle x^{a-1}e^{-bx}}$, and ${\displaystyle \mathbf {x} \mid u}$ conditionally follows ${\displaystyle N({\boldsymbol {\mu }},u^{-1}{\boldsymbol {\Sigma }})}$.

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.[2][3]

## Conditional Distribution

This was demonstrated by Muirhead [4] though previously derived using the simpler ratio representation above, by Cornish.[5] Let vector ${\displaystyle X}$ follow the multivariate t distribution and partition into two subvectors of ${\displaystyle p_{1},p_{2}}$ elements:

${\displaystyle X_{p}={\begin{bmatrix}X_{1}\\X_{2}\end{bmatrix}}\sim t_{p}\left(\mu _{p},\Sigma _{p\times p},\nu \right)}$

where ${\displaystyle p_{1}+p_{2}=p}$, the known mean vector is ${\displaystyle \mu _{p}={\begin{bmatrix}\mu _{1}\\\mu _{2}\end{bmatrix}}}$ and the scale matrix is ${\displaystyle \Sigma _{p\times p}={\begin{bmatrix}\Sigma _{11}&\Sigma _{12}\\\Sigma _{21}&\Sigma _{22}\end{bmatrix}}}$.

Then

${\displaystyle p(X_{2}|X_{1})\sim t_{p_{2}}\left(\mu _{2|1},{\frac {\nu +d_{1}}{\nu +p_{1}}}\Sigma _{22|1},\nu +p_{1}\right)}$

where

${\displaystyle \mu _{2|1}=\mu _{2}+\Sigma _{21}\Sigma _{11}^{-1}\left(X_{1}-\mu _{1}\right)}$ is the conditional mean where it exists or median otherwise.
${\displaystyle \Sigma _{22|1}=\Sigma _{22}-\Sigma _{12}\Sigma _{11}^{-1}\Sigma _{21}}$ is the Schur complement of ${\displaystyle \Sigma _{11}{\text{ in }}\Sigma .}$
${\displaystyle d_{1}=(X_{1}-\mu _{1})^{T}\Sigma _{11}^{-1}(X_{1}-\mu _{1})}$ is the squared Mahalanobis distance of ${\displaystyle X_{1}}$ from ${\displaystyle \mu _{1}}$ with scale matrix ${\displaystyle \Sigma _{11}}$

See [6] for a simple proof of the above conditional distribution.

## Copulas based on the multivariate t

The use of such distributions[7] is enjoying renewed interest due to applications in mathematical finance, especially through the use of the Student's t copula.[citation needed]

## Elliptical Representation

Constructed as an elliptical distribution[8] and in the simplest centralised case with spherical symmetry and without scaling, ${\displaystyle \Sigma =\operatorname {I} \,}$, the multivariate t PDF takes the form

${\displaystyle f_{X}(X)=g(X^{T}X)={\frac {\Gamma {\big (}{\frac {1}{2}}(\nu +p)\,{\big )}}{(\nu \pi )^{\,p/2}\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}{\bigg (}1+\nu ^{-1}X^{T}X{\bigg )}^{-(\nu +p)/2}}$

where ${\displaystyle X=(x_{1},\cdots ,x_{p})^{T}{\text{ is a sampled }}p{\text{-vector}}}$ and ${\displaystyle \nu }$ = degrees of freedom. Muirhead (section 1.5) refers to this as a multivariate Cauchy distribution. The expected covariance of ${\displaystyle X}$ is

${\displaystyle \int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(x_{1},\dots ,x_{p})XX^{T}\,dx_{1}\dots dx_{p}={\frac {\nu }{\nu -2}}\operatorname {E} (XX^{T})}$

The aim is to convert the Cartesian PDF to a radial one. Kibria and Joarder,[9] in a tutorial-style paper, define radial measure ${\displaystyle r_{2}=R^{2}={\frac {X^{T}X}{p}}}$ such that

${\displaystyle \operatorname {E} [r_{2}]=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(x_{1},\dots ,x_{p}){\frac {X^{T}X}{p}}\,dx_{1}\dots dx_{p}}$

which is equivalent to the expected variance of ${\displaystyle p}$-element vector ${\displaystyle X}$ treated as a univariate zero-mean random sequence. They note that ${\displaystyle r_{2}}$ follows the Fisher-Snedecor or ${\displaystyle F}$ distribution:

${\displaystyle r_{2}\sim F_{F}(p,\nu )=B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}{\bigg (}{\frac {p}{\nu }}{\bigg )}^{p/2}r_{2}^{p/2-1}{\bigg (}1+{\frac {p}{\nu }}r_{2}{\bigg )}^{-(p+\nu )/2}}$

having mean value ${\displaystyle \operatorname {E} [r_{2}]={\frac {\nu }{\nu -2}}}$.

By a change of random variable to ${\displaystyle y={\frac {p}{\nu }}r_{2}={\frac {X^{T}X}{\nu }}}$ in the equation above, retaining ${\displaystyle p}$-vector ${\displaystyle X}$, we have ${\displaystyle \operatorname {E} [y]=\int _{-\infty }^{\infty }\cdots \int _{-\infty }^{\infty }f_{X}(X){\frac {X^{T}X}{\nu }}\,dx_{1}\dots dx_{p}={\frac {p}{\nu -2}}}$ and probability distribution

{\displaystyle {\begin{aligned}f_{Y}(y|\,p,\nu )&={\frac {\nu }{p}}B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}{\big (}{\frac {p}{\nu }}{\big )}^{\,p/2}{\big (}{\frac {p}{\nu }}{\big )}^{-p/2-1}y^{\,p/2-1}{\big (}1+y{\big )}^{-(p+\nu )/2}\\\\&=B{\bigg (}{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}^{-1}y^{\,p/2-1}(1+y)^{-(\nu +p)/2}\end{aligned}}}

which is a regular Beta-prime distribution ${\displaystyle y\sim \beta \,'{\bigg (}y;{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}}$ having mean value ${\displaystyle {\frac {{\frac {1}{2}}p}{{\frac {1}{2}}\nu -1}}={\frac {p}{\nu -2}}}$. The cumulative distribution function of ${\displaystyle y}$ is thus known to be

${\displaystyle F_{Y}(y)\sim I\,{\bigg (}{\frac {y}{1+y}};\,{\frac {p}{2}},{\frac {\nu }{2}}{\bigg )}}$

where ${\displaystyle I}$ is the incomplete Beta function.

These results can be derived by straightforward transformation of coordinates from cartesian to spherical. A constant radius surface at ${\displaystyle R=(X^{T}X)^{1/2}}$ with PDF ${\displaystyle p_{X}(X)\propto {\bigg (}1+\nu ^{-1}R^{2}{\bigg )}^{-(\nu +p)/2}}$ is an iso-density surface. The quantum of probability in a surface shell of area ${\displaystyle A_{R}}$ and thickness ${\displaystyle \delta R}$ at ${\displaystyle R}$ is ${\displaystyle \delta P=p_{X}(R)\,A_{R}\delta R}$.

The enclosed sphere in ${\displaystyle p}$ dimensions has surface area ${\displaystyle A_{R}={\frac {2\pi ^{p/2}R^{\,p-1}}{\Gamma (p/2)}}}$ and substitution into ${\displaystyle \delta P}$ shows that the shell has element of probability ${\displaystyle \delta P=p_{X}(R){\frac {2\pi ^{p/2}R^{p-1}}{\Gamma (p/2)}}\delta R}$. This is equivalent to a radial density function

${\displaystyle f_{R}(R)={\frac {\Gamma {\big (}{\frac {1}{2}}(\nu +p)\,{\big )}}{\nu ^{\,p/2}\pi ^{\,p/2}\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}{\frac {2\pi ^{p/2}R^{p-1}}{\Gamma (p/2)}}{\bigg (}1+{\frac {R^{2}}{\nu }}{\bigg )}^{-(\nu +p)/2}}$

which simplifies to ${\displaystyle f_{R}(R)={\frac {2}{\nu ^{1/2}B{\big (}{\frac {1}{2}}p,{\frac {1}{2}}\nu {\big )}}}{\bigg (}{\frac {R^{2}}{\nu }}{\bigg )}^{(p-1)/2}{\bigg (}1+{\frac {R^{2}}{\nu }}{\bigg )}^{-(\nu +p)/2}}$ where ${\displaystyle B(*,*)}$ is the Beta function.

Changing the radial variable to ${\displaystyle y=R^{2}/\nu }$ returns the previous Beta Prime distribution ${\displaystyle f_{Y}(y)={\frac {1}{B{\big (}{\frac {1}{2}}p,{\frac {1}{2}}\nu {\big )}}}y^{\,p/2-1}{\bigg (}1+y{\bigg )}^{-(\nu +p)/2}}$

To scale the radial variables without changing the radial shape function, define scale matrix ${\displaystyle \Sigma =\alpha \operatorname {I} }$ , yielding a 3-parameter Cartesian density function, ie. the probability ${\displaystyle \Delta _{P}}$ in volume element ${\displaystyle dx_{1}\dots dx_{p}}$ is

${\displaystyle \Delta _{P}{\big (}f_{X}(X\,|\alpha ,p,\nu ){\big )}={\frac {\Gamma {\big (}{\frac {1}{2}}(\nu +p)\,{\big )}}{(\nu \pi )^{\,p/2}\alpha ^{\,p/2}\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}{\bigg (}1+{\frac {X^{T}X}{\alpha \nu }}{\bigg )}^{-(\nu +p)/2}\;dx_{1}\dots dx_{p}}$

or, in terms of scalar radial variable ${\displaystyle R}$,

${\displaystyle f_{R}(R\,|\alpha ,p,\nu )={\frac {2}{\alpha ^{1/2}\;\nu ^{1/2}B{\big (}{\frac {1}{2}}p,{\frac {1}{2}}\nu {\big )}}}{\bigg (}{\frac {R^{2}}{\alpha \,\nu }}{\bigg )}^{(p-1)/2}{\bigg (}1+{\frac {R^{2}}{\alpha \,\nu }}{\bigg )}^{-(\nu +p)/2}}$

The moments of all the radial variables can be derived from the Beta Prime distribution. If ${\displaystyle Z\sim \beta '(a,b)}$ then ${\displaystyle \operatorname {E} (Z^{m})={\frac {B(a+m,b-m)}{B(a,b)}}}$, a known result. Thus, for variable ${\displaystyle y}$, proportional to ${\displaystyle R^{2}}$, we have

${\displaystyle \operatorname {E} (y^{m})={\frac {B({\frac {1}{2}}p+m,{\frac {1}{2}}\nu -m)}{B({\frac {1}{2}}p,{\frac {1}{2}}\nu )}}={\frac {\Gamma {\big (}{\frac {1}{2}}p+m{\big )}\;\Gamma {\big (}{\frac {1}{2}}\nu -m{\big )}}{\Gamma {\big (}{\frac {1}{2}}p{\big )}\;\Gamma {\big (}{\frac {1}{2}}\nu {\big )}}}}$

The moments of ${\displaystyle r_{2}=\nu \,y}$ are

${\displaystyle \operatorname {E} (r_{2}^{m})=\nu ^{m}\operatorname {E} (y^{m})}$

while introducing the scale matrix ${\displaystyle \alpha \operatorname {I} }$ yields

${\displaystyle \operatorname {E} (r_{2}^{m}|\alpha )=\alpha ^{m}\nu ^{m}\operatorname {E} (y^{m})}$

Moments relating to radial variable ${\displaystyle R}$ are found by setting ${\displaystyle R=(\alpha \nu y)^{1/2}}$ and ${\displaystyle M=2m}$ whereupon

${\displaystyle \operatorname {E} (R^{M})=\operatorname {E} {\big (}(\alpha \nu y)^{1/2}{\big )}^{2m}=(\alpha \nu )^{M/2}\operatorname {E} (y^{M/2})=(\alpha \nu )^{M/2}{\frac {B{\big (}{\frac {1}{2}}(p+M),{\frac {1}{2}}(\nu -M){\big )}}{B({\frac {1}{2}}p,{\frac {1}{2}}\nu )}}}$

## Linear Combinations and Affine Transformation

Following section 3.3 of Kibria et.al. let ${\displaystyle Z}$ be a ${\displaystyle p}$-vector sampled from a central spherical multivariate t distribution with ${\displaystyle \nu }$ degrees of freedom: ${\displaystyle Z_{p}\sim t_{p}(0,\operatorname {I} ,\nu )}$. ${\displaystyle X}$ is derived from ${\displaystyle Z}$ via a linear transformation:

${\displaystyle X=\mu +\Sigma ^{1/2}Z}$

where ${\displaystyle \Sigma }$ has full rank, then

${\displaystyle X\sim t_{p}(\mu ,\Sigma ,\nu )}$

That is ${\displaystyle \operatorname {E} (X)=\mu }$ and the covariance of ${\displaystyle X}$ is ${\displaystyle \operatorname {E} {\big [}(X-\mu )(X-\mu )^{T}{\big ]}={\frac {\nu }{\nu -2}}\Sigma }$

Furthermore, if ${\displaystyle A}$ is a non-singular matrix then

${\displaystyle Y=AX+b}$ ${\displaystyle \sim t_{p}(A\mu +b,A\Sigma A^{T},\nu )}$

with mean ${\displaystyle \operatorname {E} (Y)=A\mu +b}$ and covariance ${\displaystyle \operatorname {E} {\big [}(Y-A\mu -b)(Y-A\mu -b)^{T}{\big ]}={\frac {\nu }{\nu -2}}A\Sigma A^{T}}$.

Roth (reference below) notes that if ${\displaystyle A}$ is a ${\displaystyle s\times p}$ squat matrix with ${\displaystyle s then ${\displaystyle Y}$ has distribution ${\displaystyle Y_{s}\sim t_{s}(A\mu +b,A\Sigma A^{T},\nu )}$.

If ${\displaystyle A}$ takes the form ${\displaystyle Y_{s}={\begin{bmatrix}\operatorname {I_{s\times s}} &0_{s\times (p-s)}\end{bmatrix}}X_{p}}$ then the PDF of ${\displaystyle Y_{s}}$ is the marginal distribution of the leading ${\displaystyle s}$ elements of ${\displaystyle X_{p}}$.

In the above, the degrees of freedom parameter ${\displaystyle \nu }$ remains invariant throughout and all vectors must ultimately derive from one initial isotropic spherical vector ${\displaystyle Z}$ whose elements are not statistically independent. Adding two sample multivariate t vectors generated with independent Chi-squared samples and different ${\displaystyle \nu }$ values: ${\textstyle {1}/{\sqrt {u_{1}/\nu _{1}}},\;\;{1}/{\sqrt {u_{2}/\nu _{2}}}}$ , as defined in the leading paragraph, will not produce internally consistent distributions, though they will yield a Behrens-Fisher problem.[10]

## 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.

## References

1. ^ Roth, Michael (17 April 2013). "On the Multivariate t Distribution" (PDF). Automatic Control group. Linköpin University, Sweden. Archived (PDF) from the original on 31 July 2022. Retrieved 1 June 2022.
2. ^ Botev, Z. I.; L'Ecuyer, P. (6 December 2015). "Efficient probability estimation and simulation of the truncated multivariate student-t distribution". 2015 Winter Simulation Conference (WSC). Huntington Beach, CA, USA: IEEE. pp. 380–391. doi:10.1109/WSC.2015.7408180.
3. ^ Genz, Alan (2009). Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics. Vol. 195. Springer. doi:10.1007/978-3-642-01689-9. ISBN 978-3-642-01689-9. Archived from the original on 2022-08-27. Retrieved 2017-09-05.
4. ^ Muirhead, Robb (1982). Aspects of Multivariate Statistical Theory. USA: Wiley. pp. 32-36 Theorem 1.5.4. ISBN 978-0-47 1-76985-9.
5. ^ Cornish, E A (1954). "The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates". Australian Journal of Physics. 7: 531–542. doi:10.1071/PH550193.
6. ^ Ding, Peng (2016). "On the Conditional Distribution of the Multivariate t Distribution". The American Statistician. 70 (3): 293-295. arXiv:1604.00561. doi:10.1080/00031305.2016.1164756. S2CID 55842994.
7. ^ Demarta, Stefano; McNeil, Alexander (2004). "The t Copula and Related Copulas" (PDF). Risknet.
8. ^ Osiewalski, Jacek; Steele, Mark (1996). Bayesian Analysis in Statistics and Econometrics Ch(27): Posterior Moments of Scale Parameters in Elliptical Sampling Models. Wiley. pp. 323–335. ISBN 0-471-11856-7.
9. ^ Kibria, K M G; Joarder, A H (Jan 2006). "A short review of multivariate t distribution" (PDF). Journal of Statistical Research. 40 (1): 59–72. doi:10.1007/s42979-021-00503-0. S2CID 232163198.
10. ^ Giron, Javier; del Castilo, Carmen (2010). "The multivariate Behrens–Fisher distribution". Journal of Multivariate Analysis. 101 (9): 2091–2102. doi:10.1016/j.jmva.2010.04.008.

## 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.