Matrix variate beta distribution

Matrix variate beta distribution
Notation
Parameters
Support	matrices with both and positive definite
PDF
CDF

In statistics, the matrix variate beta distribution is a generalization of the beta distribution. 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 $\Sigma$ .

References

A. K. Gupta and D. K. Nagar 1999. "Matrix variate distributions". Chapman and Hall.
S. K. Mitra 1970. "A density-free approach to matrix variate beta distribution". The Indian Journal of Statistics, Series A, (1961-2002), volume 32, number 1 (March 1970), pp81-88.

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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	${}_{1}F_{1}\left(a;a+b;iZ\right)$