Matrix t-distribution

Matrix t
Notation	${\rm T}_{n,p}(\nu,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)$
Parameters	$\mathbf{M}$ location (real $n\times p$ matrix) $\boldsymbol\Omega$ scale (positive-definite real $p\times p$ matrix) $\boldsymbol\Sigma$ scale (positive-definite real $n\times n$ matrix) $\nu$ degrees of freedom
Support	$\mathbf{X} \in\mathbb{R}^{n\times p}$
PDF	$\frac{\Gamma_p\left(\frac{\nu+n+p-1}{2}\right)}{(\pi)^\frac{np}{2} \Gamma_p\left(\frac{\nu+p-1}{2}\right)} \|\boldsymbol\Omega\|^{-\frac{n}{2}} \|\boldsymbol\Sigma\|^{-\frac{p}{2}}$ $\times \left\|\mathbf{I}_n + \boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right\|^{-\frac{\nu+n+p-1}{2}}$
CDF	No analytic expression
Mean	$\mathbf{M}$ if $\nu + p - n > 1$ , else undefined
Mode	$\mathbf{M}$
Variance	$\frac{\boldsymbol\Sigma \otimes \boldsymbol\Omega}{\nu-2}$ 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.^[1] 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 $\mathbf{X}$ of an $n\times p$ space is

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}},

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 $\Gamma_p$ 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	${\rm T}_{n,p}(\alpha,\beta,\mathbf{M},\boldsymbol\Sigma, \boldsymbol\Omega)$
Parameters	$\mathbf{M}$ location (real $n\times p$ matrix) $\boldsymbol\Omega$ scale (positive-definite real $p\times p$ matrix) $\boldsymbol\Sigma$ 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	$\frac{\Gamma_p(\alpha+n/2)}{(2\pi/\beta)^\frac{np}{2} \Gamma_p(\alpha)} \|\boldsymbol\Omega\|^{-\frac{n}{2}} \|\boldsymbol\Sigma\|^{-\frac{p}{2}}$ $\times \left\|\mathbf{I}_n + \frac{\beta}{2}\boldsymbol\Sigma^{-1}(\mathbf{X} - \mathbf{M})\boldsymbol\Omega^{-1}(\mathbf{X}-\mathbf{M})^{\rm T}\right\|^{-(\alpha+n/2)}$ $\Gamma_p$ is the multivariate gamma function.
CDF	No analytic expression
Mean	$\mathbf{M}$
Variance	$\frac{2(\boldsymbol\Sigma \otimes \boldsymbol\Omega)}{\beta(2\alpha-n-1)}$
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.^[2]

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

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

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

This makes use of the following:

\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) =

\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) .

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

\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}) .

The characteristic function is^[2]

\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),

where

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

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

Notes

↑ Zhu, Shenghuo and Kai Yu and Yihong Gong (2007). "Predictive Matrix-Variate t Models." In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, NIPS '07: Advances in Neural Information Processing Systems 20, pages 1721–1728. MIT Press, Cambridge, MA, 2008. The notation is changed a bit in this article for consistency with the matrix normal distribution article.
1 2 Iranmanesh, Anis, M. Arashi and S. M. M. Tabatabaey (2010). "On Conditional Applications of Matrix Variate Normal Distribution". Iranian Journal of Mathematical Sciences and Informatics, 5:2, pp. 33–43.

External links

A C++ library for random matrix generator

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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