In statistics, the matrix variate beta distribution is a generalization of the beta distribution. 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:
Matrix variate beta distributionNotation | |
---|
Parameters | |
---|
Support | matrices with both and positive definite |
---|
PDF | |
---|
CDF | |
---|
Here is the multivariate beta function:
where is the multivariate gamma function given by
Theorems
Distribution of matrix inverse
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 .
Partitioned matrix results
If and we partition as
where is and is , then defining the Schur complement as gives the following results:
- is independent of
- has an inverted matrix variate t distribution, specifically
Wishart 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 .
See also
References
- Gupta, A. K.; Nagar, D. K. (1999). Matrix Variate Distributions. Chapman and Hall. ISBN 1-58488-046-5.
- 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.