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 is a positive definite matrix with a matrix variate beta distribution, and are real parameters, we write (sometimes ). The probability density function for is:
Here is the multivariate beta function:
where is the multivariate gamma function given by
If then the density of is given by
provided that and .
If and is a constant orthogonal matrix, then
Also, if is a random orthogonal matrix which is independent of , then , distributed independently of .
If is any constant , matrix of rank , then has a generalized matrix variate beta distribution, specifically .
If and we partition as
where is and is , then defining the Schur complement as gives the following results:
Mitra proves the following theorem which illustrates a useful property of the matrix variate beta distribution. Suppose are independent Wishart matrices . Assume that is positive definite and that . If
where , then has a matrix variate beta distribution . In particular, is independent of .
The spectral density is expressed by a Jacobi polynomial.
The distribution of the largest eigenvalue is well approximated by a transform of the TracyâÂÂWidom distribution.