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InverseWishartMatrixDistribution

InverseWishartMatrixDistribution

InverseWishartMatrixDistribution[ν,Σ]

represents an inverse Wishart matrix distribution with ν degrees of freedom and covariance matrix Σ.

Details

The probability density for a symmetric matrix in an inverse Wishart matrix distribution is proportional to , where is the size of matrix Σ.
For a matrix distributed as InverseWishartMatrixDistribution[ν,Σ], the inverse is distributed as WishartMatrixDistribution[ν,Σ^-1].
The covariance matrix can be any positive definite symmetric matrix of dimensions and ν can be any real number greater than .
InverseWishartMatrixDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

Examples

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Basic Examples (3)

Generate a pseudorandom matrix:

Check that it is positive definite:

Sample eigenvalues of an inverse Wishart random matrix using MatrixPropertyDistribution:

Mean and variance:

Scope (6)

Generate a single pseudorandom matrix:

Generate a set of pseudorandom matrices:

Sample at extended precision:

Compute statistical properties numerically:

Numerically approximate expectation of the largest matrix eigenvalue :

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood for both distributions:

Skewness:

Properties & Relations (3)

, where and are independent Gaussian vector and Wishart matrix follows HotellingTSquareDistribution:

Use MatrixPropertyDistribution to sample expressions :

Any diagonal element of inverse Wishart random matrix follows scaled inverse χ² distribution:

Diagonal elements are not independent:

For any nonzero vector and inverse Wishart matrix with scale matrix , is χ² distributed:

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