matrix t-distribution

For a matrix t-distribution, the probability density function at the point $\backslash mathbf\{X\}$ of an $n\backslash times\; p$ space is

:$f(\backslash mathbf\{X\}\; ;\; \backslash nu,\backslash mathbf\{M\},\backslash boldsymbol\backslash Sigma,\; \backslash boldsymbol\backslash Omega)\; =\; K$

\times \left|\mathbf{I}_n + \boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-\frac{\nu+n+p-1}{2}},

where the constant of integration K is given by

:$K\; =$

\frac{\Gamma_p\left(\frac{\nu+n+p-1}{2}\right)}{(\pi)^\frac{np}{2} \Gamma_p\left(\frac{\nu+p-1}{2}\right)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}}.

Here $\backslash Gamma\_p$ is the multivariate gamma function.

If $\backslash mathbf\{X\}\; \backslash sim\; \backslash mathcal\{T\}\_\{n\backslash times\; p\}(\backslash nu,\; \backslash mathbf\{M\},\; \backslash mathbf\{\backslash Sigma\},\; \backslash mathbf\{\backslash Omega\})$, then we have the following properties:

The mean, or expected value is, if $\backslash nu\; >\; 1$:

:$E[\backslash mathbf\{X\}]\; =\; \backslash mathbf\{M\}$

and we have the following second-order expectations, if $\backslash nu\; >\; 2$:

:$E[(\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})(\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})^\{T\}]$

= \frac{\mathbf{\Sigma}\operatorname{tr}(\mathbf{\Omega})}{\nu-2}

:$E[(\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})^\{T\}\; (\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})]$

= \frac{\mathbf{\Omega}\operatorname{tr}(\mathbf{\Sigma}) }{\nu-2}

where $\backslash operatorname\{tr\}$ denotes trace.

More generally, for appropriately dimensioned matrices A,B,C:

:$\backslash begin\{align\}$

E[(\mathbf{X}- \mathbf{M})\mathbf{A}(\mathbf{X}- \mathbf{M})^{T}]

&= \frac{\mathbf{\Sigma}\operatorname{tr}(\mathbf{A}^T\mathbf{\Omega})}{\nu - 2} \\

E[(\mathbf{X}- \mathbf{M})^T\mathbf{B}(\mathbf{X}- \mathbf{M})]

&= \frac{\mathbf{\Omega}\operatorname{tr}(\mathbf{B}^T \mathbf{\Sigma})}{\nu - 2} \\

E[(\mathbf{X}- \mathbf{M})\mathbf{C}(\mathbf{X}- \mathbf{M})]

&= \frac{\mathbf{\Sigma}\mathbf{C}^T\mathbf{\Omega}}{\nu - 2}

\end{align}

Transpose transform:

:$\backslash mathbf\{X\}^T\; \backslash sim\backslash mathcal\{T\}\_\{p\backslash times\; n\}(\backslash nu,\; \backslash mathbf\{M\}^T,\; \backslash mathbf\{\backslash Omega\},\; \backslash mathbf\{\backslash Sigma\})$

Linear transform: let A (r-by-n), be of full rank r ≤ n and B (p-by-s), be of full rank s ≤ p, then:

:$\backslash mathbf\{AXB\}\backslash sim\; \backslash mathcal\{T\}\_\{r\backslash times\; s\}(\backslash nu,\backslash mathbf\{AMB\},\; \backslash mathbf\{A\backslash Sigma\; A\}^T,\; \backslash mathbf\{B\}^T\backslash mathbf\{\backslash Omega\; B\}\; )$

The characteristic function and various other properties can be derived from the re-parameterised formulation (see below).

{{Probability distribution|

name =Re-parameterized matrix t|

type =density|

pdf_image =|

cdf_image =|

notation =$\{\backslash rm\; T\}\_\{n,p\}(\backslash alpha,\backslash beta,\backslash mathbf\{M\},\backslash boldsymbol\backslash Sigma,\; \backslash boldsymbol\backslash Omega)$|

parameters =

$\backslash mathbf\{M\}$ location (real $n\backslash times\; p$ matrix)

$\backslash boldsymbol\backslash Omega$ scale (positive-definite real $p\backslash times\; p$ matrix)

$\backslash boldsymbol\backslash Sigma$ scale (positive-definite real $n\backslash times\; n$ matrix)

$\backslash alpha\; >\; (p-1)/2$ shape parameter

$\backslash beta\; >\; 0$ scale parameter |

support =$\backslash mathbf\{X\}\; \backslash in\backslash mathbb\{R\}^\{n\backslash times\; p\}$|

pdf =$\backslash frac\{\backslash Gamma\_p(\backslash alpha+n/2)\}\{(2\backslash pi/\backslash beta)^\backslash frac\{np\}\{2\}\; \backslash Gamma\_p(\backslash alpha)\}\; |\backslash boldsymbol\backslash Omega|^\{-\backslash frac\{n\}\{2\}\}\; |\backslash boldsymbol\backslash Sigma|^\{-\backslash frac\{p\}\{2\}\}$

:$\backslash times\; \backslash left|\backslash mathbf\{I\}\_n\; +\; \backslash frac\{\backslash beta\}\{2\}\backslash boldsymbol\backslash Sigma^\{-1\}(\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})\backslash boldsymbol\backslash Omega^\{-1\}(\backslash mathbf\{X\}-\backslash mathbf\{M\})^\{\backslash rm\; T\}\backslash right|^\{-(\backslash alpha+n/2)\}$

• $\backslash Gamma\_p$ is the multivariate gamma function.

|

cdf =No analytic expression|

mean =$\backslash mathbf\{M\}$ if $\backslash alpha\; >\; p/2$, else undefined|

median =|

mode =|

variance =$\backslash frac\{2(\backslash boldsymbol\backslash Sigma\; \backslash otimes\; \backslash boldsymbol\backslash Omega)\}\{\backslash beta(2\backslash alpha-p-1)\}$ if $\backslash alpha\; >\; (p+1)/2$, else undefined|

skewness =|

kurtosis =|

entropy =|

mgf =|

char =see below|

}}

An alternative parameterisation of the matrix t-distribution uses two parameters $\backslash alpha$ and $\backslash beta$ in place of $\backslash nu$.Iranmanesh, Anis, M. Arashi and S. M. M. Tabatabaey (2010). [http://www.ijmsi.ir/browse.php?a_id=139&slc_lang=en&sid=1&ftxt=1 "On Conditional Applications of Matrix Variate Normal Distribution"]. Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.

This formulation reduces to the standard matrix t-distribution with $\backslash beta=2,\; \backslash alpha=\backslash frac\{\backslash nu+p-1\}\{2\}.$

This formulation of the matrix t-distribution can be derived as the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse multivariate gamma distribution placed over either of its covariance matrices.

If $\backslash mathbf\{X\}\; \backslash sim\; \{\backslash rm\; T\}\_\{n,p\}(\backslash alpha,\backslash beta,\backslash mathbf\{M\},\backslash boldsymbol\backslash Sigma,\; \backslash boldsymbol\backslash Omega)$ then

:$\backslash mathbf\{X\}^\{\backslash rm\; T\}\; \backslash sim\; \{\backslash rm\; T\}\_\{p,n\}(\backslash alpha,\backslash beta,\backslash mathbf\{M\}^\{\backslash rm\; T\},\backslash boldsymbol\backslash Omega,\; \backslash boldsymbol\backslash Sigma).$

The property above comes from Sylvester's determinant theorem:

:$\backslash det\backslash left(\backslash mathbf\{I\}\_n\; +\; \backslash frac\{\backslash beta\}\{2\}\backslash boldsymbol\backslash Sigma^\{-1\}(\backslash mathbf\{X\}\; -\; \backslash mathbf\{M\})\backslash boldsymbol\backslash Omega^\{-1\}(\backslash mathbf\{X\}-\backslash mathbf\{M\})^\{\backslash rm\; T\}\backslash right)\; =$

::$\backslash det\backslash left(\backslash mathbf\{I\}\_p\; +\; \backslash frac\{\backslash beta\}\{2\}\backslash boldsymbol\backslash Omega^\{-1\}(\backslash mathbf\{X\}^\{\backslash rm\; T\}\; -\; \backslash mathbf\{M\}^\{\backslash rm\; T\})\backslash boldsymbol\backslash Sigma^\{-1\}(\backslash mathbf\{X\}^\{\backslash rm\; T\}-\backslash mathbf\{M\}^\{\backslash rm\; T\})^\{\backslash rm\; T\}\backslash right)\; .$

If $\backslash mathbf\{X\}\; \backslash sim\; \{\backslash rm\; T\}\_\{n,p\}(\backslash alpha,\backslash beta,\backslash mathbf\{M\},\backslash boldsymbol\backslash Sigma,\; \backslash boldsymbol\backslash Omega)$ and $\backslash mathbf\{A\}(n\backslash times\; n)$ and $\backslash mathbf\{B\}(p\backslash times\; p)$ are nonsingular matrices then

:$\backslash mathbf\{AXB\}\; \backslash sim\; \{\backslash rm\; T\}\_\{n,p\}(\backslash alpha,\backslash beta,\backslash mathbf\{AMB\},\backslash mathbf\{A\}\backslash boldsymbol\backslash Sigma\backslash mathbf\{A\}^\{\backslash rm\; T\},\; \backslash mathbf\{B\}^\{\backslash rm\; T\}\backslash boldsymbol\backslash Omega\backslash mathbf\{B\})$

.

The characteristic function is

:$\backslash phi\_T(\backslash mathbf\{Z\})\; =\; \backslash frac\{\backslash exp(\{\backslash rm\; tr\}(i\backslash mathbf\{Z\}\text{'}\backslash mathbf\{M\}))|\backslash boldsymbol\backslash Omega|^\backslash alpha\}\{\backslash Gamma\_p(\backslash alpha)(2\backslash beta)^\{\backslash alpha\; p\}\}\; |\backslash mathbf\{Z\}\text{'}\backslash boldsymbol\backslash Sigma\backslash mathbf\{Z\}|^\backslash alpha\; B\_\backslash alpha\backslash left(\backslash frac\{1\}\{2\backslash beta\}\backslash mathbf\{Z\}\text{'}\backslash boldsymbol\backslash Sigma\backslash mathbf\{Z\}\backslash boldsymbol\backslash Omega\backslash right),$

where

:$B\_\backslash delta(\backslash mathbf\{WZ\})\; =\; |\backslash mathbf\{W\}|^\{-\backslash delta\}\; \backslash int\_\{\backslash mathbf\{S\}>0\}\; \backslash exp\backslash left(\{\backslash rm\; tr\}(-\backslash mathbf\{SW\}-\backslash mathbf\{S^\{-1\}Z\})\backslash right)|\backslash mathbf\{S\}|^\{-\backslash delta-\backslash frac12(p+1)\}d\backslash mathbf\{S\},$

and where $B\_\backslash delta$ is the type-two Bessel function of Herz{{clarify|date=April 2016}} of a matrix argument.

• Multivariate t-distribution
• Matrix normal distribution

{{Reflist}}

• [https://github.com/zweng/rmg A C++ library for random matrix generator]

{{ProbDistributions|multivariate}}

Category:Random matrices

Category:Multivariate continuous distributions