Matrix t-distribution

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

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

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 $𝐗\sim {𝒯}_{n\times p}(\nu ,𝐌,\mathrm{\Sigma },\mathrm{\Omega })$, 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 $\nu >2$:

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

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

where $\mathrm{tr}$ denotes trace.

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

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

Transformation

Transpose transform:

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

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:

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

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

Re-parameterized matrix t-distribution

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

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

The property above comes from Sylvester's determinant theorem:

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

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

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

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

The characteristic function is

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

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 HerzTemplate:Clarify of a matrix argument.

See also

• Multivariate t-distribution
• Matrix normal distribution

External links

• A C++ library for random matrix generator

Template:ProbDistributions