Matrix variate beta distribution

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.

Matrix variate beta distribution
Notation	${\rm {B}}_{p}(a,b)$
Parameters	$a,b$
Support	$p\times p$ matrices with both $U$ and $I_{p}-U$ positive definite
PDF	$\left\{\beta _{p}\left(a,b\right)\right\}^{-1}\det \left(U\right)^{a-(p+1)/2}\det \left(I_{p}-U\right)^{b-(p+1)/2}.$
CDF	${}_{1}F_{1}\left(a;a+b;iZ\right)$

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.

Extreme value distribution

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

See also

• Matrix variate Dirichlet distribution

References

• 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.