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Matrix variate beta distribution

Generalization of beta distribution From Wikipedia, the free encyclopedia

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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.

Quick Facts Notation, Parameters ...

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

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Theorems

Distribution of matrix inverse

If then the density of is given by

provided that and .

Orthogonal transform

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 .

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Spectral density

The spectral density is expressed by a Jacobi polynomial.[1]

Extreme value distribution

The distribution of the largest eigenvalue is well approximated by a transform of the Tracy–Widom distribution.[2]

See also

References

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