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.

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 ${\displaystyle U\sim B_p(a,b)}$ (sometimes ${\displaystyle B_p^I(a,b)}$). The probability density function for ${\displaystyle U}$ is:$${\displaystyle {\left\{{\beta }_p(a,b)\right\}}^{-1}\det {\left(U\right)}^{a-(p+1)/2}\det {(I_p-U)}^{b-(p+1)/2}.}$$

Here ${\displaystyle {\beta }_p(a,b)}$ is the multivariate beta function:

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

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

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

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

${\displaystyle \frac{1}{{\beta }_p(a,b)}\det (X{)}^{-(a+b)}\det {(X-I_p)}^{b-(p+1)/2}}$

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

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

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

If ${\displaystyle A}$ is any constant ${\displaystyle q\times p}$, ${\displaystyle q\le p}$ matrix of rank ${\displaystyle q}$, then ${\displaystyle AU{A}^T}$ has a generalized matrix variate beta distribution, specifically ${\displaystyle AU{A}^T\sim G{B}_q(a,b;A{A}^T,0)}$.

If ${\displaystyle U\sim B_p(a,b)}$ and we partition ${\displaystyle U}$ as

${\displaystyle U=\left[\begin{array}{cc}{U}_{11}& U_{12}\\ U_{21}& U_{22}\end{array}\right]}$

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 ${\displaystyle U_{22\cdot 1}}$
• ${\displaystyle U_{11}\sim B_{p_1}(a,b)}$
• ${\displaystyle U_{22\cdot 1}\sim B_{p_2}(a-p_1/2,b)}$
• ${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}}$ has an inverted matrix variate t distribution, specifically ${\displaystyle U_{21}\mid U_{11},U_{22\cdot 1}\sim I{T}_{p_2,p_1}(2b-p+1,0,I_{p_2}-U_{22\cdot 1},U_{11}(I_{p_1}-U_{11})).}$

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 ${\displaystyle S_1\sim W_p(n_1,\mathrm{\Sigma }),S_2\sim W_p(n_2,\mathrm{\Sigma })}$. Assume that ${\displaystyle \mathrm{\Sigma }}$ is positive definite and that ${\displaystyle n_1+n_2\ge p}$. 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 ${\displaystyle \mathrm{\Sigma }}$.

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

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

• Matrix variate Dirichlet distribution

• 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.. 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. 

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