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{{Short description|Multivariate continuous probability distribution }}
{{Orphan|date=December 2023}}
{{Probability distribution |
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support =<math>\mathbf{X}</math> is ''p'' × ''p'' [[positive-definite matrix|positive definite matrix]]|
pdf =<math>
\frac{\Gamma_p\left(\frac{\nu+\delta+p-1}{2}\right)}{\Gamma_p\left(\frac{\nu}{2}\right)\
</math>
*<math>\Gamma_p</math> is the [[multivariate gamma function]]
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}}
In [[statistics]], the '''matrix F distribution''' (or '''matrix variate F distribution''') is a matrix variate generalization of the [[F-distribution|F distribution]] which is defined on real-valued [[positive-definite matrix|positive-definite]] [[matrix (mathematics)|matrices]]. In [[Bayesian statistics]] it can be used as the semi conjugate prior for the covariance matrix or precision matrix of [[multivariate normal]] distributions, and related distributions.<ref name="olkinrubin1964">{{Cite journal |last1=Olkin |first1=Ingram |last2=Rubin |first2=Herman |date=1964-03-01 |title=Multivariate Beta Distributions and Independence Properties of the Wishart Distribution |url=http://projecteuclid.org/euclid.aoms/1177703748 |journal=The Annals of Mathematical Statistics |language=en |volume=35 |issue=1 |pages=261–269 |doi=10.1214/aoms/1177703748 |issn=0003-4851|doi-access=free }}</ref><ref name="dawid1981">{{Cite journal |last=Dawid |first=A. P. |date=1981 |title=Some matrix-variate distribution theory: Notational considerations and a Bayesian application |url=https://academic.oup.com/biomet/article-lookup/doi/10.1093/biomet/68.1.265 |journal=Biometrika |language=en |volume=68 |issue=1 |pages=265–274 |doi=10.1093/biomet/68.1.265 |issn=0006-3444|url-access=subscription }}</ref><ref name="mulderpericchi2018">{{Cite journal |last1=Mulder |first1=Joris |last2=Pericchi |first2=Luis Raúl |date=2018-12-01 |title=The Matrix-F Prior for Estimating and Testing Covariance Matrices
==Density==
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<math>
f_{\mathbf X}({\mathbf X}; {\mathbf \Psi}, \nu, \delta) =
\frac{\Gamma_p\left(\frac{\nu+\delta+p-1}{2}\right)}{\Gamma_p\left(\frac{\nu}{2}\right)\
</math>
where <math>\mathbf{X}</math> and <math>{\mathbf\Psi}</math> are <math>p\times p</math> [[positive-definite matrix|positive definite]] matrices, <math>| \cdot |</math> is the determinant, Γ<sub>''p''</sub>(&
==Properties==
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and <math>{\mathbf \Phi_2}\sim \mathcal{W}({\mathbf I_p},\delta+k-1)</math>, and define <math>\mathbf X = {\mathbf \Phi_2}^{-1/2}{\mathbf \Phi_1}{\mathbf \Phi_2}^{-1/2}</math>, then <math>\mathbf X\sim \mathcal{F}({\mathbf I_p},\nu,\delta) </math>.
* If <math>{\mathbf X}|\mathbf\Phi\sim \mathcal{W}^{-1}({\mathbf\Phi},\delta+p-1)</math> and <math>{\mathbf \Phi}\sim \mathcal{W}({\mathbf\Psi},\nu)</math>, then, after integrating out <math>\mathbf\Phi</math>, <math>\mathbf X</math> has a matrix F-distribution, i.e.,
<math>
f_{\mathbf X | \mathbf\Phi, \nu, \delta}(\mathbf X) =
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== Related distributions ==
* The matrix F-distribution has also been termed the multivariate beta II distribution.<ref name="tan1969">{{Cite journal |last=Tan |first=W. Y. |date=1969-03-01 |title=Note on the Multivariate and the Generalized Multivariate Beta Distributions |url=http://www.tandfonline.com/doi/abs/10.1080/01621459.1969.10500966 |journal=Journal of the American Statistical Association |language=en |volume=64 |issue=325 |pages=230–241 |doi=10.1080/01621459.1969.10500966 |issn=0162-1459|url-access=subscription }}</ref> See also,<ref name="perez2017">{{Cite journal |last1=Pérez |first1=María-Eglée |last2=Pericchi |first2=Luis Raúl |last3=Ramírez |first3=Isabel Cristina |date=2017-09-01 |title=The Scaled Beta2 Distribution as a Robust Prior for Scales
* A [[univariate]] version of the matrix F distribution is the [[F-distribution]]. With <math>p=1</math> (i.e. univariate) and <math>\mathbf\Psi = 1</math>, and <math>x=\mathbf{X}</math>, the [[probability density function]] of the matrix F distribution becomes the univariate (unscaled) [[F-distribution|F distribution]]:<br/><math>
f_{x\mid\nu, \delta}(x) =
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</math>
* In the [[univariate]] case, with <math>p=1</math> and <math>x=\mathbf{X}</math>, and when setting <math>\nu=1</math>, then <math>\sqrt{x}</math> follows a [[Folded-t and half-t distributions|half t distribution]] with scale parameter <math>\sqrt{\psi}</math> and degrees of freedom <math>\delta</math>. The half t distribution is a common prior for standard deviations<ref name="gelman2006">{{Cite journal |last=Gelman |first=Andrew |date=2006-09-01 |title=Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper)
==See also==
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