Matrix variate beta distribution

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

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

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

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

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

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

Theorems

Distribution of matrix inverse

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

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

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

Orthogonal transform

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

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

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

Partitioned matrix results

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

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

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

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

Wishart results

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

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

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

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]

See also

• Matrix variate Dirichlet distribution

References

1. ( Potters & Bouchaud 2020 )
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). 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.