CDF No analytic expression | ||
Notation T n , p ( ν , M , Σ , Ω ) {\displaystyle {\rm {T}}_{n,p}(\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})} Parameters M {\displaystyle \mathbf {M} } location (real n × p {\displaystyle n\times p} matrix) Ω {\displaystyle {\boldsymbol {\Omega }}} scale (positive-definite real p × p {\displaystyle p\times p} matrix) Σ {\displaystyle {\boldsymbol {\Sigma }}} scale (positive-definite real n × n {\displaystyle n\times n} matrix) ν {\displaystyle \nu } degrees of freedom Support X ∈ R n × p {\displaystyle \mathbf {X} \in \mathbb {R} ^{n\times p}} PDF Γ p ( ν + n + p − 1 2 ) ( π ) n p 2 Γ p ( ν + p − 1 2 ) | Ω | − n 2 | Σ | − p 2 {\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}}}} × | I n + Σ − 1 ( X − M ) Ω − 1 ( X − M ) T | − ν + n + p − 1 2 {\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}}}} Mean M {\displaystyle \mathbf {M} } if ν + p − n > 1 {\displaystyle \nu +p-n>1} , else undefined |
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.
Contents
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
where the constant of integration K is given by
Here
The characteristic function and various other properties can be derived from the generalized matrix t-distribution (see below).
Generalized matrix t-distribution
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
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
This makes use of the following:
If
The characteristic function is
where
and where