Harman Patil (Editor)

Matrix t distribution

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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 X of an n × p space is

f ( X ; ν , M , Σ , Ω ) = K × | I n + Σ − 1 ( X − M ) Ω − 1 ( X − M ) T | − ν + n + p − 1 2 ,

where the constant of integration K is given by

K = Γ p ( ν + n + p − 1 2 ) ( π ) n p 2 Γ p ( ν + p − 1 2 ) | Ω | − n 2 | Σ | − p 2 .

Here Γ 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

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 β = 2 , α = ν + 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 X ∼ T n , p ( α , β , M , Σ , Ω ) then

X T ∼ T p , n ( α , β , M T , Ω , Σ ) .

This makes use of the following:

det ( I n + β 2 Σ − 1 ( X − M ) Ω − 1 ( X − M ) T ) = det ( I p + β 2 Ω − 1 ( X T − M T ) Σ − 1 ( X T − M T ) T ) .

If X ∼ T n , p ( α , β , M , Σ , Ω ) and A ( n × n ) and B ( p × p ) are nonsingular matrices then

A X B ∼ T n , p ( α , β , A M B , A Σ A T , B T Ω B ) .

The characteristic function is

ϕ T ( Z ) = exp ⁡ ( t r ( i Z ′ M ) ) | Ω | α Γ p ( α ) ( 2 β ) α p | Z ′ Σ Z | α B α ( 1 2 β Z ′ Σ Z Ω ) ,

where

B δ ( W Z ) = | W | − δ ∫ S > 0 exp ⁡ ( t r ( − S W − S − 1 Z ) ) | S | − δ − 1 2 ( p + 1 ) d S ,

and where B δ is the type-two Bessel function of Herz of a matrix argument.

References

Matrix t-distribution Wikipedia
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