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Matrix variate beta distribution

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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 distribution
Notation
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

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Distribution of matrix inverse

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If then the density of is given by

provided that and .

Orthogonal transform

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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

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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

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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

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References

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  • 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.