Matrix t-distribution

Matrix t

Notation	[math]\displaystyle{ {\rm T}_{n,p}(\nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega) }[/math]
Parameters	[math]\displaystyle{ \mathbf{M} }[/math] location (real [math]\displaystyle{ n\times p }[/math] matrix) [math]\displaystyle{ \boldsymbol\Omega }[/math] scale (positive-definite real [math]\displaystyle{ p\times p }[/math] matrix) [math]\displaystyle{ \boldsymbol\Sigma }[/math] scale (positive-definite real [math]\displaystyle{ n\times n }[/math] matrix) [math]\displaystyle{ \nu }[/math] degrees of freedom
Support	[math]\displaystyle{ \mathbf{X} \in\mathbb{R}^{n\times p} }[/math]
PDF	[math]\displaystyle{ \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}} }[/math] [math]\displaystyle{ \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}} }[/math]
CDF	No analytic expression
Mean	[math]\displaystyle{ \mathbf{M} }[/math] if [math]\displaystyle{ \nu + p - n \gt 1 }[/math], else undefined
Mode	[math]\displaystyle{ \mathbf{M} }[/math]
Variance	[math]\displaystyle{ \frac{\boldsymbol\Sigma \otimes \boldsymbol\Omega}{\nu-2} }[/math] if [math]\displaystyle{ \nu \gt 2 }[/math], 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. For example, the matrix t-distribution is the compound distribution that results from sampling from a matrix normal distribution having sampled the covariance matrix of the matrix normal from an inverse Wishart distribution.

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 [math]\displaystyle{ \mathbf{X} }[/math] of an [math]\displaystyle{ n\times p }[/math] space is

[math]\displaystyle{ f(\mathbf{X} ; \nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\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}}, }[/math]

where the constant of integration K is given by

[math]\displaystyle{ 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}}. }[/math]

Here [math]\displaystyle{ \Gamma_p }[/math] is the multivariate gamma function.

The characteristic function and various other properties can be derived from the generalized matrix t-distribution (see below).

Generalized matrix t-distribution

Generalized matrix t

Notation	[math]\displaystyle{ {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega) }[/math]
Parameters	[math]\displaystyle{ \mathbf{M} }[/math] location (real [math]\displaystyle{ n\times p }[/math] matrix) [math]\displaystyle{ \boldsymbol\Omega }[/math] scale (positive-definite real [math]\displaystyle{ p\times p }[/math] matrix) [math]\displaystyle{ \boldsymbol\Sigma }[/math] scale (positive-definite real [math]\displaystyle{ n\times n }[/math] matrix) [math]\displaystyle{ \alpha \gt (p-1)/2 }[/math] shape parameter [math]\displaystyle{ \beta \gt 0 }[/math] scale parameter
Support	[math]\displaystyle{ \mathbf{X} \in\mathbb{R}^{n\times p} }[/math]
PDF	[math]\displaystyle{ \frac{\Gamma_p(\alpha+n/2)}{(2\pi/\beta)^\frac{np}{2} \Gamma_p(\alpha)} |\boldsymbol\Omega|^{-\frac{n}{2}} |\boldsymbol\Sigma|^{-\frac{p}{2}} }[/math] [math]\displaystyle{ \times \left|\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right|^{-(\alpha+n/2)} }[/math] [math]\displaystyle{ \Gamma_p }[/math] is the multivariate gamma function.
CDF	No analytic expression
Mean	[math]\displaystyle{ \mathbf{M} }[/math]
Variance	[math]\displaystyle{ \frac{2(\boldsymbol\Sigma \otimes \boldsymbol\Omega)}{\beta(2\alpha-p-1)} }[/math]
CF	see below

The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters α and β in place of ν.

This reduces to the standard matrix t-distribution with [math]\displaystyle{ \beta=2, \alpha=\frac{\nu+p-1}{2}. }[/math]

The generalized matrix t-distribution is 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 [math]\displaystyle{ \mathbf{X} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega) }[/math] then

[math]\displaystyle{ \mathbf{X}^{\rm T} \sim {\rm T}_{p,n}(\alpha,\beta,\mathbf{M}^{\rm T},\boldsymbol\Omega, \boldsymbol\Sigma). }[/math]

The property above comes from Sylvester's determinant theorem:

[math]\displaystyle{ \det\left(\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right) = }[/math]

[math]\displaystyle{ \det\left(\mathbf{I}_p + \frac{\beta}{2}\boldsymbol\Omega^{-1}(\mathbf{X}^{\rm T} - \mathbf{M}^{\rm T})\boldsymbol\Sigma^{-1}(\mathbf{X}^{\rm T}-\mathbf{M}^{\rm T})^{\rm T}\right) . }[/math]

If [math]\displaystyle{ \mathbf{X} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega) }[/math] and [math]\displaystyle{ \mathbf{A}(n\times n) }[/math] and [math]\displaystyle{ \mathbf{B}(p\times p) }[/math] are nonsingular matrices then

[math]\displaystyle{ \mathbf{AXB} \sim {\rm T}_{n,p}(\alpha,\beta,\mathbf{AMB},\mathbf{A}\boldsymbol\Sigma\mathbf{A}^{\rm T}, \mathbf{B}^{\rm T}\boldsymbol\Omega\mathbf{B}) . }[/math]

The characteristic function is

[math]\displaystyle{ \phi_T(\mathbf{Z}) = \frac{\exp({\rm tr}(i\mathbf{Z}'\mathbf{M}))|\boldsymbol\Omega|^\alpha}{\Gamma_p(\alpha)(2\beta)^{\alpha p}} |\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}|^\alpha B_\alpha\left(\frac{1}{2\beta}\mathbf{Z}'\boldsymbol\Sigma\mathbf{Z}\boldsymbol\Omega\right), }[/math]

where

[math]\displaystyle{ B_\delta(\mathbf{WZ}) = |\mathbf{W}|^{-\delta} \int_{\mathbf{S}\gt 0} \exp\left({\rm tr}(-\mathbf{SW}-\mathbf{S^{-1}Z})\right)|\mathbf{S}|^{-\delta-\frac12(p+1)}d\mathbf{S}, }[/math]

and where [math]\displaystyle{ B_\delta }[/math] 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