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In statistics, the matrix variate beta distribution is a generalization of the beta distribution. It is also called the MANOVA ensemble and the Jacobi ensemble.

If $U$ is a $p\times p$ positive definite matrix with a matrix variate beta distribution, and $a,b>(p-1)/2$ are real parameters, we write $U\sim B_{p}\left(a,b\right)$ (sometimes $B_{p}^{I}\left(a,b\right)$ ). The probability density function for $U$ is: $\left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.$

Here $\beta _{p}\left(a,b\right)$ is the multivariate beta function:

\beta _{p}\left(a,b\right)={\frac {\Gamma _{p}\left(a\right)\Gamma _{p}\left(b\right)}{\Gamma _{p}\left(a+b\right)}}

where $\Gamma _{p}\left(a\right)$ is the multivariate gamma function given by

\Gamma _{p}\left(a\right)=\pi ^{p(p-1)/4}\prod _{i=1}^{p}\Gamma \left(a-(i-1)/2\right).

Theorems

Distribution of matrix inverse

If $U\sim B_{p}(a,b)$ then the density of $X=U^{-1}$ is given by

{\frac {1}{\beta _{p}\left(a,b\right)}}\det(X)^{-(a+b)}\det \left(X-I_{p}\right)^{b-(p+1)/2}

provided that $X>I_{p}$ and $a,b>(p-1)/2$ .

Orthogonal transform

If $U\sim B_{p}(a,b)$ and $H$ is a constant $p\times p$ orthogonal matrix, then $HUH^{T}\sim B(a,b).$

Also, if $H$ is a random orthogonal $p\times p$ matrix which is independent of $U$ , then $HUH^{T}\sim B_{p}(a,b)$ , distributed independently of $H$ .

If $A$ is any constant $q\times p$ , $q\leq p$ matrix of rank $q$ , then $AUA^{T}$ has a generalized matrix variate beta distribution, specifically $AUA^{T}\sim GB_{q}\left(a,b;AA^{T},0\right)$ .

Partitioned matrix results

If $U\sim B_{p}\left(a,b\right)$ and we partition $U$ as

U={\begin{bmatrix}U_{11}&U_{12}\\U_{21}&U_{22}\end{bmatrix}}

where $U_{11}$ is $p_{1}\times p_{1}$ and $U_{22}$ is $p_{2}\times p_{2}$ , then defining the Schur complement $U_{22\cdot 1}$ as $U_{22}-U_{21}{U_{11}}^{-1}U_{12}$ gives the following results:

$U_{11}$ is independent of $U_{22\cdot 1}$
$U_{11}\sim B_{p_{1}}\left(a,b\right)$
$U_{22\cdot 1}\sim B_{p_{2}}\left(a-p_{1}/2,b\right)$
$U_{21}\mid U_{11},U_{22\cdot 1}$ has an inverted matrix variate t distribution, specifically $U_{21}\mid U_{11},U_{22\cdot 1}\sim IT_{p_{2},p_{1}}\left(2b-p+1,0,I_{p_{2}}-U_{22\cdot 1},U_{11}(I_{p_{1}}-U_{11})\right).$

Wishart results

Mitra proves the following theorem which illustrates a useful property of the matrix variate beta distribution. Suppose $S_{1},S_{2}$ are independent Wishart $p\times p$ matrices $S_{1}\sim W_{p}(n_{1},\Sigma ),S_{2}\sim W_{p}(n_{2},\Sigma )$ . Assume that $\Sigma$ is positive definite and that $n_{1}+n_{2}\geq p$ . If

U=S^{-1/2}S_{1}\left(S^{-1/2}\right)^{T},

where $S=S_{1}+S_{2}$ , then $U$ has a matrix variate beta distribution $B_{p}(n_{1}/2,n_{2}/2)$ . In particular, $U$ is independent of $\Sigma$ .

Spectral density

The spectral density is expressed by a Jacobi polynomial.^[1]

Extreme value distribution

The distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution.^[2]

References

^ (Potters & Bouchaud 2020)
^ Johnstone, Iain M. (2008-12-01). “Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence”. The Annals of Statistics. 36 (6). arXiv:0803.3408. doi:10.1214/08-AOS605. ISSN 0090-5364.

Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30). “7. The Jacobi Ensemble”. A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists. Cambridge University Press. doi:10.1017/9781108768900. ISBN 978-1-108-76890-0.
Forrester, Peter (2010). “3. Laguerre and Jacobi ensembles”. Log-gases and random matrices. London Mathematical Society monographs. Princeton: Princeton University Press. ISBN 978-0-691-12829-0.
Anderson, G.W.; Guionnet, A.; Zeitouni, O. (2010). “4. Some generalities”. An introduction to random matrices. Cambridge: Cambridge University Press. ISBN 978-0-521-19452-5.
Mehta, M.L. (2004). “19. Matrix ensembles and classical orthogonal polynomials”. Random Matrices. Amsterdam: Elsevier/Academic Press. ISBN 0-12-088409-7.
Gupta, A. K.; Nagar, D. K. (1999). Matrix Variate Distributions. Chapman and Hall. ISBN 1-58488-046-5.
Khatri, C. G. (1992). “Matrix Beta Distribution with Applications to Linear Models, Testing, Skewness and Kurtosis”. In Venugopal, N. (ed.). Contributions to Stochastics. John Wiley & Sons. pp. 26–34. ISBN 0-470-22050-3.
Mitra, S. K. (1970). “A density-free approach to matrix variate beta distribution”. The Indian Journal of Statistics. Series A (1961–2002). 32 (1): 81–88. JSTOR 25049638.

[rdp-we-cite_note-1] (Potters & Bouchaud 2020)

[rdp-we-cite_note-2] Johnstone, Iain M. (2008-12-01). “Multivariate analysis and Jacobi ensembles: Largest eigenvalue, Tracy–Widom limits and rates of convergence”. The Annals of Statistics. 36 (6). arXiv:0803.3408. doi:10.1214/08-AOS605. ISSN 0090-5364.

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[2]

Random matrix theory
Concepts	Ensemble Spectrum Universality Resolvent Level repulsion Integrability Free probability Noncrossing partition Coulomb gas Dyson Brownian motion Riemann–Hilbert problem Determinantal point process
Ensembles	Gaussian ensemble Wishart ensemble Jacobi ensemble Ginibre ensemble Beta ensemble Circular ensemble Deformed ensemble Matrix t-distribution Random band ensemble Heavy-tailed
Laws	Wigner semicircle distribution Marchenko–Pastur law Circular law Tracy–Widom distribution BBP transition Wigner surmise
Techniques	Stieltjes transformation Isserlis’s theorem Fredholm determinant Orthogonal polynomials Skew-orthogonal polynomials Christoffel–Darboux formula Cavity method Weingarten function Selberg integral Mean-field theory Airy process Bessel process sine process Painlevé transcendents KPZ equation Green’s function

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