Matrix F-distribution: Difference between revisions

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{{Short description|TheMultivariate page describes acontinuous probability distribution that is currently not included in Wikipedia.}}

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{{Probability distribution |

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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, I. and|first1=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|doi-access=free }}</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/68.1.265 265–274.|issn=0006-3444|url-access=subscription }}</ref><ref name="mulderpericchi2018">{{Cite journal |last1=Mulder, J. and|first1=Joris |last2=Pericchi, L.|first2=Luis R.Raúl (2010). [https://projecteuclid.org/journals/bayesian|date=2018-analysis/volume12-13/issue-4/The-Matrix-F-Prior-for-Estimating-and-Testing-Covariance-Matrices/10.1214/17-BA1092.full01 "|title=The Matrix-F Prior for Estimating and Testing Covariance Matrices"]. ''|journal=Bayesian Analysis'', |volume=13: |issue=4, pp|doi=10.1214/17-BA1092 1193|s2cid=126398943 |issn=1936-1214.0975|doi-access=free }}</ref><ref name="williamsmulder2020">{{Cite journal |last1=Williams, D.|first1=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"]. ''|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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===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>

f_{\mathbf X | \mathbf\Phi, \nu, \delta}(\mathbf X) =

\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 [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|journal=Journal onof the multivariate and the generalized multivariate beta distributions."]. ''Journal of American Statistical Association'', |language=en |volume=64, pp.|issue=325 |pages=230–241 |doi=10.1080/01621459.1969.10500966 |issn=0162-1459|url-access=subscription }}</ref>. See also ,<ref name="perez2017">Perez,{{Cite M.-E.journal and|last1=Pérez |first1=María-Eglée |last2=Pericchi, L.|first2=Luis R.Raúl and|last3=Ramírez Ramirez,|first3=Isabel I.Cristina C. (|date=2017). [https://projecteuclid.org/journals/bayesian-analysis/volume09-12/issue-3/The-Scaled-Beta2-Distribution-as-a-Robust-Prior-for-Scales/10.1214/16-BA1015.short01 "|title=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|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) =

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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 A.|first=Andrew (|date=2006). [https://projecteuclid.org/journals/bayesian-analysis/volume09-1/issue-3/Prior-distributions-for-variance-parameters-in-hierarchical-models-comment-on/10.1214/06-BA117A.full01 "|title=Prior distributions for variance parameters in hierarchical models."]. ''(comment on article by Browne and Draper) |journal=Bayesian Analysis'', |volume=1: |issue=3, pp|doi=10.1214/06-BA117A 515–534.|issn=1936-0975|doi-access=free }}</ref>.

==See also==

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{{ProbDistributions|multivariate}}

[[Category:Analysis of variance]]

[[Category:Multivariate continuous distributions]]