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]
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Matrix tNotation |
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location (real matrix)
scale (positive-definite real matrix)
scale (positive-definite real matrix)
degrees of freedom (real) |
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if , else undefined |
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if , else undefined |
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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]
For a matrix t-distribution, the probability density function at the point of an space is
- ;\nu ,\mathbf {M} ,{\boldsymbol {\Sigma }},{\boldsymbol {\Omega }})=K\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}}},}
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).
Quick Facts Notation, Parameters ...
Generalized matrix tNotation |
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location (real matrix)
scale (positive-definite real matrix)
scale (positive-definite real matrix)
shape parameter
scale parameter |
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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.
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.
Gupta, Arjun K and Nagar, Daya K (1999). Matrix variate distributions. CRC Press. pp. Chapter 4.{{cite book}}
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