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Matrix t-distribution

From Wikipedia, the free encyclopedia

Matrix t
Notation
Parameters location (real matrix)

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

degrees of freedom
Support
PDF
CDFNo analytic expression
Mean if , else undefined
Mode
Variance if , else undefined
CFsee 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.[clarification needed] 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.[citation needed]

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 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

Generalized matrix t
Notation
Parameters location (real matrix)

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

scale parameter
Support
PDF
CDFNo analytic expression
Mean
Variance
CFsee 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

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 then[citation needed]

The property above comes from Sylvester's determinant theorem:

If and and are nonsingular matrices then[citation needed]

The characteristic function is[2]

where

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

See also

Notes

  1. ^ 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 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


This page was last edited on 6 August 2019, at 22:42
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