Matrix F-distribution: Difference between revisions

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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}}</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}}</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 |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 |journal=Bayesian Analysis |volume=13 |issue=4 |doi=10.1214/17-BA1092 |s2cid=126398943 |issn=1936-0975}}</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 |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|s2cid=225019695 }}</ref>.

==Density==

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===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" />. 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>.

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\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>

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\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===

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<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>

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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}}</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 |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.full |journal=Bayesian Analysis |volume=12 |issue=3 |doi=10.1214/16-BA1015 |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>