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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 |last=Olkin, I. and|first=Ingram |last2=Rubin, H.|first2=Herman (|date=1964). [https://projecteuclid.org/journals/annals-of03-mathematical-statistics/volume-35/issue-1/Multivariate-Beta-Distributions-and-Independence-Properties-of-the-Wishart-Distribution/10.1214/aoms/1177703748.full01 "|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, pp.|issue=1 |pages=261–269 |doi=10.1214/aoms/1177703748 |issn=0003-4851}}</ref><ref name="dawid1981">{{Cite journal |last=Dawid, |first=A. P. (|date=1981). [https://academic.oup.com/biomet/article-abstract/68/1/265/237681?login|title=true "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, pp. |pages=265–274 |doi=10.1093/biomet/68.1.265 |issn=0006-3444}}</ref><ref name="mulderpericchi2018">Mulder,{{Cite J.journal and|last=Mulder |first=Joris |last2=Pericchi, L.|first2=Luis R.Raúl (2010).|date=2018-12-01 [|title=The Matrix-F Prior for Estimating and Testing Covariance Matrices |url=https://projecteuclid.org/journals/bayesian-analysis/volume-13/issue-4/The-Matrix-F-Prior-for-Estimating-and-Testing-Covariance-Matrices/10.1214/17-BA1092.full "The Matrix-F Prior for Estimating and Testing Covariance Matrices"]. ''|journal=Bayesian Analysis'', |volume=13: |issue=4, pp|doi=10.1214/17-BA1092 1193|issn=1936-1214.0975}}</ref><ref name="williamsmulder2020">{{Cite journal |last=Williams, D.|first=Donald R. and |last2=Mulder, J.|first2=Joris (|date=2020).-12-01 [https://www.sciencedirect.com/science/article/pii/S0022249620300821 "|title=Bayesian hypothesis testing for Gaussian graphical models: Conditional independence and order constraints"] |url=https://linkinghub.elsevier.com/retrieve/pii/S0022249620300821 ''|journal=Journal of Mathematical Psychology'', |language=en |volume=99, |pages=102441 |doi=10.1016/j.jmp.2020.102441}}</ref>.
==Density==
===Construction of the distribution===
* The standard matrix F distribution, with an identity scale matrix <math>\mathbf I_p</math>, was originally derived by <ref name="olkinrubin1964">< /ref>. When considering independent distributions, <math>{\mathbf \Phi_1}\sim \mathcal{W}({\mathbf I_p},\nu)</math> 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.,<br/><math>
\int f_{\mathbf X | \mathbf\Phi, \delta+p-1}(\mathbf X)
f_{\mathbf\Phi | \mathbf\Psi, \nu}(\mathbf\Phi) d\mathbf\Phi.
</math> <br/>This construction is useful to construct a semi-conjugate prior for a covariance matrix<ref name="mulderpericchi2018" />.
*If <math>{\mathbf X}|\mathbf\Phi\sim \mathcal{W}({\mathbf\Phi},\nu)</math> and <math>{\mathbf \Phi}\sim \mathcal{W}^{-1}({\mathbf\Psi},\delta+p-1)</math>, then, after integrating out <math>\mathbf\Phi</math>, <math>\mathbf X</math> has a matrix F-distribution, i.e.,<br/><math>
\int f_{\mathbf X | \mathbf\Phi, \nu}(\mathbf X)
f_{\mathbf\Phi | \mathbf\Psi, \delta + p - 1}(\mathbf\Phi) d\mathbf\Phi.
</math><br/>This construction is useful to construct a semi-conjugate prior for a precision matrix<ref name="williamsmulder2020" />.
===Marginal distributions from a matrix F distributed matrix===
<math> E(\mathbf X) = \frac{\nu}{\delta-2}\mathbf\Psi.</math>
The (co)variance of elements of <math>\mathbf{X}</math> are given by<ref name="mulderpericchi2018" />:
:<math>
== 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 [https|title=Note on the Multivariate and the Generalized Multivariate Beta Distributions |url=http://www.tandfonline.com/doi/abs/10.1080/01621459.1969.10500966 "Note on the multivariate and the generalized multivariate beta distributions."]. ''|journal=Journal of the American Statistical Association'', |language=en |volume=64, pp.|issue=325 |pages=230–241 |doi=10.1080/01621459.1969.10500966 |issn=0162-1459}}</ref>. See also <ref name="perez2017">Perez,{{Cite M.-E.journal and|last=Pérez |first=María-Eglée |last2=Pericchi, L.|first2=Luis R.Raúl and|last3=Ramírez Ramirez,|first3=Isabel I.Cristina C. (|date=2017).-09-01 [|title=The Scaled Beta2 Distribution as a Robust Prior for Scales |url=https://projecteuclid.org/journals/bayesian-analysis/volume-12/issue-3/The-Scaled-Beta2-Distribution-as-a-Robust-Prior-for-Scales/10.1214/16-BA1015.shortfull "The Scaled Beta2 Distribution as a Robust Prior for Scales."]. ''|journal=Bayesian Analysis'', |volume=12: |issue=3, pp|doi=10.1214/16-BA1015 615–637.|issn=1936-0975}}</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>
</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 A.|first=Andrew (|date=2006-09-01 |title=Prior distributions for variance parameters in hierarchical models (comment on article by Browne and Draper). [|url=https://projecteuclid.org/journals/bayesian-analysis/volume-1/issue-3/Prior-distributions-for-variance-parameters-in-hierarchical-models-comment-on/10.1214/06-BA117A.full "Prior distributions for variance parameters in hierarchical models."]. ''|journal=Bayesian Analysis'', |volume=1: |issue=3, pp|doi=10.1214/06-BA117A 515–534.|issn=1936-0975}}</ref>.
==See also==
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