Matrix t-distribution

Short description: Concept in statistics

Matrix t

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

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

where the constant of integration K is given by

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

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

Properties

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

Expected values

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

$E[\mathbf{𝐗}]=\mathbf{𝐌}$

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

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

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

where $\mathrm{tr}$ denotes trace.

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

$\begin{array}{rl}E[(\mathbf{𝐗}-\mathbf{𝐌})\mathbf{𝐀}(\mathbf{𝐗}-\mathbf{𝐌}{)}^T]& =\frac{\mathrm{\Sigma }\mathrm{tr}({\mathbf{𝐀}}^T\mathrm{\Omega })}{\nu -2}\\ E[(\mathbf{𝐗}-\mathbf{𝐌}{)}^T\mathbf{𝐁}(\mathbf{𝐗}-\mathbf{𝐌})]& =\frac{\mathrm{\Omega }\mathrm{tr}({\mathbf{𝐁}}^T\mathrm{\Sigma })}{\nu -2}\\ E[(\mathbf{𝐗}-\mathbf{𝐌})\mathbf{𝐂}(\mathbf{𝐗}-\mathbf{𝐌})]& =\frac{\mathrm{\Sigma }{\mathbf{𝐂}}^T\mathrm{\Omega }}{\nu -2}\end{array}$

Transformation

Transpose transform:

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

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:

$\mathbf{𝐀}\mathbf{𝐗}\mathbf{𝐁}\sim {{𝒯}}_{r\times s}(\nu ,\mathbf{𝐀}\mathbf{𝐌}\mathbf{𝐁},{\mathbf{𝐀}\mathrm{\Sigma }\mathbf{𝐀}}^T,{\mathbf{𝐁}}^T\mathrm{\Omega }\mathbf{𝐁})$

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

${\mathbf{𝐗}}^{\mathrm{T}}\sim {\mathrm{T}}_{p,n}(\alpha ,\beta ,{\mathbf{𝐌}}^{\mathrm{T}},\mathrm{\Omega },\mathrm{\Sigma }).$

The property above comes from Sylvester's determinant theorem:

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

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

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

$\mathbf{𝐀}\mathbf{𝐗}\mathbf{𝐁}\sim {\mathrm{T}}_{n,p}(\alpha ,\beta ,\mathbf{𝐀}\mathbf{𝐌}\mathbf{𝐁},\mathbf{𝐀}\mathrm{\Sigma }{\mathbf{𝐀}}^{\mathrm{T}},{\mathbf{𝐁}}^{\mathrm{T}}\mathrm{\Omega }\mathbf{𝐁}).$

The characteristic function is

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

where

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

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

Template:Random matrix theory