Matrix F-distribution: Difference between revisions

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support =<math>\mathbf{X}</math> is ''p''&nbsp;×&nbsp;''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)\Gamma_kGamma_p\left(\frac{\delta+p-1}{2}\right)|\mathbf{\Psi}|^{\frac{\nu}{2}}}~|{\mathbf X}|^{\frac{\nu-p-1}{2}} |\textbf{I}_p+{\mathbf X}\mathbf{\Psi}^{-1}|^{-\frac{\nu+\delta+p-1}{2}}

</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 |journal=Bayesian Analysis |volume=13 |issue=4 |doi=10.1214/17-BA1092 |s2cid=126398943 |issn=1936-0975|doi-access=free }}</ref><ref name="williamsmulder2020">{{Cite journal |last1=Williams |first1=Donald R. |last2=Mulder |first2=Joris |date=2020-12-01 |title=Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints |journal=Journal of Mathematical Psychology |language=en |volume=99 |pages=102441 |doi=10.1016/j.jmp.2020.102441|s2cid=225019695 |doi-access=free }}</ref>

==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)\Gamma_kGamma_p\left(\frac{\delta+p-1}{2}\right)|\mathbf{\Psi}|^{\frac{\nu}{2}}}~|{\mathbf X}|^{\frac{\nu-p-1}{2}} |\textbf{I}_p+{\mathbf X}\mathbf{\Psi}^{-1}|^{-\frac{\nu+\delta+p-1}{2}}

</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, &Gamma;_''p''(&middotsdot;) is the [[multivariate gamma function]], and <math>\textbf{I}_p</math> is the ''p''&nbsp;×&nbsp;''p'' [[identity matrix]].

==Properties==

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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 |journal=Bayesian Analysis |volume=12 |issue=3 |doi=10.1214/16-BA1015 |issn=1936-0975|doi-access=free }}</ref> for a univariate version.

* 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) =