Matrix variate beta distribution

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.