Jump to content

Matrix t-distribution

From Wikipedia, the free encyclopedia
(Redirected from Matrix t distribution)
Matrix t
Notation
Parameters

location (real matrix)
scale (positive-definite real matrix)
scale (positive-definite real matrix)

degrees of freedom (real)
Support
PDF

CDF No analytic expression
Mean if , else undefined
Mode
Variance if , 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][2]

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,[1] and the multivariate t-distribution can be generated in a similar way.[2]

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.[3]

Definition

[edit]

For a matrix t-distribution, the probability density function at the point of an space is

where the constant of integration K is given by

Here 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

[edit]
Generalized matrix t
Notation
Parameters

location (real matrix)
scale (positive-definite real matrix)
scale (positive-definite real matrix)
shape parameter

scale parameter
Support
PDF

CDF No analytic expression
Mean if , else undefined
Variance if , else undefined
CF see below

The generalized matrix t-distribution is a generalization of the matrix t-distribution with two parameters and in place of .[3]

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

[edit]

If then[2][3]

The property above comes from Sylvester's determinant theorem:

If and and are nonsingular matrices then[2][3]

The characteristic function is[3]

where

and where is the type-two Bessel function of Herz[clarification needed] of a matrix argument.

See also

[edit]

Notes

[edit]
  1. ^ a b 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.
  2. ^ a b c d Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.{{cite book}}: CS1 maint: multiple names: authors list (link)
  3. ^ a b c d e 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.
[edit]