Matrix t-distribution

Matrix t
Notation	${\mathrm{T}}_{n,p}(\nu ,𝐌,𝜮,𝜴)$
Parameters	$𝐌$ location (real $n\times p$ matrix) $𝜴$ scale (positive-definite real $n\times n$ matrix) $𝜮$ scale (positive-definite real $p\times p$ matrix) $\nu >0$ degrees of freedom (real)
Support	$𝐗\in {ℝ}^{n\times p}$
PDF	$\frac{{\Gamma }_p\left(\frac{\nu +n+p-1}{2}\right)}{(\pi {)}^{\frac{np}{2}}{\Gamma }_p\left(\frac{\nu +p-1}{2}\right)}|𝜴{|}^{-\frac{n}{2}}|𝜮{|}^{-\frac{p}{2}}$ $\times {|{𝐈}_p+{𝜮}^{-1}(𝐗-𝐌){𝜴}^{-1}(𝐗-𝐌{)}^{\mathrm{T}}|}^{-\frac{\nu +n+p-1}{2}}$
CDF	No analytic expression
Mean	$𝐌$ if $\nu >1$, else undefined
Mode	$𝐌$
Variance	$\mathrm{c}\mathrm{o}\mathrm{v}(\mathrm{v}\mathrm{e}\mathrm{c}(𝐗))=\frac{𝜮\otimes 𝜴}{\nu -2}$ if $\nu >2$, else undefined
CF	see below

In statistics, the matrix t-distribution (or matrix variate t-distribution) is the generalization of the multivariate t-distribution from vectors to matrices.

The matrix t-distribution shares the same relationship with the multivariate t-distribution that the matrix normal distribution shares with the multivariate normal distribution: If the matrix has only one row, or only one column, the distributions become equivalent to the corresponding (vector-)multivariate distribution. The matrix t-distribution is the compound distribution that results from an infinite mixture of a matrix normal distribution with an inverse Wishart distribution placed over either of its covariance matrices, and the multivariate t-distribution can be generated in a similar way.

In a Bayesian analysis of a multivariate linear regression model based on the matrix normal distribution, the matrix t-distribution is the posterior predictive distribution.

Definition

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

$f(𝐗;\nu ,𝐌,𝜮,𝜴)=K\times {|{𝐈}_n+{𝜮}^{-1}(𝐗-𝐌){𝜴}^{-1}(𝐗-𝐌{)}^{\mathrm{T}}|}^{-\frac{\nu +n+p-1}{2}},$

where the constant of integration K is given by

$K=\frac{{\Gamma }_p\left(\frac{\nu +n+p-1}{2}\right)}{(\pi {)}^{\frac{np}{2}}{\Gamma }_p\left(\frac{\nu +p-1}{2}\right)}|𝜴{|}^{-\frac{n}{2}}|𝜮{|}^{-\frac{p}{2}}.$

Here ${\Gamma }_p$ is the multivariate gamma function.

Properties

If $𝐗\sim {𝒯}_{n\times p}(\nu ,𝐌,𝜮,𝜴)$, then we have the following properties:

Expected values

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

$E[𝐗]=𝐌$

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

$E[(𝐗-𝐌)(𝐗-𝐌{)}^T]=\frac{𝜮\mathrm{tr}(𝜴)}{\nu -2}$

$E[(𝐗-𝐌{)}^T(𝐗-𝐌)]=\frac{𝜴\mathrm{tr}(𝜮)}{\nu -2}$

where $\mathrm{tr}$ denotes trace.

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

$\begin{array}{cc}E[(𝐗-𝐌)𝐀(𝐗-𝐌{)}^T]& =\frac{𝜮\mathrm{tr}({𝐀}^T𝜴)}{\nu -2}\\ E[(𝐗-𝐌{)}^T𝐁(𝐗-𝐌)]& =\frac{𝜴\mathrm{tr}({𝐁}^T𝜮)}{\nu -2}\\ E[(𝐗-𝐌)𝐂(𝐗-𝐌)]& =\frac{𝜮{𝐂}^T𝜴}{\nu -2}\end{array}$

Transformation

Transpose transform:

${𝐗}^T\sim {𝒯}_{p\times n}(\nu ,{𝐌}^T,𝜴,𝜮)$

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:

$𝐀𝐗𝐁\sim {𝒯}_{r\times s}(\nu ,𝐀𝐌𝐁,{𝐀𝜮𝐀}^T,{𝐁}^T𝜴𝐁)$

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

Re-parameterized matrix t-distribution

Re-parameterized matrix t
Notation	${\mathrm{T}}_{n,p}(\alpha ,\beta ,𝐌,𝜮,𝜴)$
Parameters	$𝐌$ location (real $n\times p$ matrix) $𝜴$ scale (positive-definite real $p\times p$ matrix) $𝜮$ scale (positive-definite real $n\times n$ matrix) $\alpha >(p-1)/2$ shape parameter $\beta >0$ scale parameter
Support	$𝐗\in {ℝ}^{n\times p}$
PDF	$\frac{{\Gamma }_p(\alpha +n/2)}{(2\pi /\beta {)}^{\frac{np}{2}}{\Gamma }_p(\alpha )}|𝜴{|}^{-\frac{n}{2}}|𝜮{|}^{-\frac{p}{2}}$ $\times {|{𝐈}_n+\frac{\beta }{2}{𝜮}^{-1}(𝐗-𝐌){𝜴}^{-1}(𝐗-𝐌{)}^{\mathrm{T}}|}^{-(\alpha +n/2)}$ ${\Gamma }_p$ is the multivariate gamma function.
CDF	No analytic expression
Mean	$𝐌$ if $\alpha >p/2$, else undefined
Variance	$\frac{2(𝜮\otimes 𝜴)}{\beta (2\alpha -p-1)}$ if $\alpha >(p+1)/2$, else undefined
CF	see below

An alternative parameterisation of the matrix t-distribution uses two parameters $\alpha$ and $\beta$ in place of $\nu$.

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

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.

Properties

If $𝐗\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,𝐌,𝜮,𝜴)$ then

${𝐗}^{\mathrm{T}}\sim {\mathrm{T}}_{p,n}(\alpha ,\beta ,{𝐌}^{\mathrm{T}},𝜴,𝜮).$

The property above comes from Sylvester's determinant theorem:

$\det ({𝐈}_n+\frac{\beta }{2}{𝜮}^{-1}(𝐗-𝐌){𝜴}^{-1}(𝐗-𝐌{)}^{\mathrm{T}})=$

$\det ({𝐈}_p+\frac{\beta }{2}{𝜴}^{-1}({𝐗}^{\mathrm{T}}-{𝐌}^{\mathrm{T}}){𝜮}^{-1}({𝐗}^{\mathrm{T}}-{𝐌}^{\mathrm{T}}{)}^{\mathrm{T}}).$

If $𝐗\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,𝐌,𝜮,𝜴)$ and $𝐀(n\times n)$ and $𝐁(p\times p)$ are nonsingular matrices then

$𝐀𝐗𝐁\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,𝐀𝐌𝐁,𝐀𝜮{𝐀}^{\mathrm{T}},{𝐁}^{\mathrm{T}}𝜴𝐁).$

The characteristic function is

${\varphi }_T(𝐙)=\frac{\exp (\mathrm{t}\mathrm{r}(i{𝐙}^{\prime }𝐌))|𝜴{|}^{\alpha }}{{\Gamma }_p(\alpha )(2\beta {)}^{\alpha p}}|{𝐙}^{\prime }𝜮𝐙{|}^{\alpha }{B}_{\alpha }\left(\frac{1}{2\beta }{𝐙}^{\prime }𝜮𝐙𝜴\right),$

where

$B_{\delta }(𝐖𝐙)=|𝐖{|}^{-\delta }\underset{𝐒>0}{\int }\exp (\mathrm{t}\mathrm{r}(-𝐒𝐖-{𝐒}^{\mathbf{-}\mathrm{𝟏}}𝐙))|𝐒{|}^{-\delta -\frac{1}{2}(p+1)}d𝐒,$

and where $B_{\delta }$ is the type-two Bessel function of Herz of a matrix argument.

See also

• Multivariate t-distribution
• Matrix normal distribution

External links

• A C++ library for random matrix generator