Matrix tNotation | |
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Parameters | location (real matrix) scale (positive-definite real matrix) scale (positive-definite real matrix) degrees of freedom (real) |
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Support | |
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PDF |
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CDF | No analytic expression |
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Mean | if , else undefined |
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Mode | |
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Variance | if , else undefined |
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CF | see below |
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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
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 tNotation | |
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Parameters | location (real matrix) scale (positive-definite real matrix) scale (positive-definite real matrix) shape parameter scale parameter |
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Support | |
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PDF | |
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CDF | No analytic expression |
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Mean | if , else undefined |
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Variance | if , else undefined |
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CF | see below |
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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
If then[2][3]
The property above comes from Sylvester's determinant theorem:
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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
Notes
- ^ 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.
- ^ 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) - ^ 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.
External links
- A C++ library for random matrix generator
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