Matrix variate beta distribution

Matrix variate beta distribution
Notation	${\mathrm{B}}_p(a,b)$
Parameters	$a,b$
Support	$p\times p$ matrices with both $U$ and $I_p-U$ positive definite
PDF	${\left\{{\beta }_p(a,b)\right\}}^{-1}\det {\left(U\right)}^{a-(p+1)/2}\det {(I_p-U)}^{b-(p+1)/2}.$
CDF	${}_1{F}_1(a;a+b;iZ)$

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

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

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

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

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

Theorems

Distribution of matrix inverse

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

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

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 $HU{H}^T\sim B(a,b).$

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

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

Partitioned matrix results

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

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

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

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 $n_1+n_2\ge p$. 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