# WishartMatrixDistribution

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

# Details

• WishartMatrixDistribution is the distribution of the sample covariance from ν independent realizations of a multivariate Gaussian distribution with covariance matrix Σ when the degrees of freedom parameter ν is an integer.
• WishartMatrixDistribution is also known as WishartLaguerre ensemble.
• The probability density for a symmetric matrix in a Wishart matrix distribution is proportional to , where is the size of matrix Σ.
• The covariance matrix can be any positive definite symmetric matrix of dimensions and ν can be any real number greater than .
• WishartMatrixDistribution 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 symmetric and positive definite:

Sample eigenvalues of a Wishart random matrix using MatrixPropertyDistribution:

Estimate joint distribution of eigenvalues:

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 and kurtosis:

## Applications(2)

When n and p (the dimension of the covariance matrix Σ) are both large, the scaled largest eigenvalue of a matrix from a Wishart ensemble with identity covariance is approximately distributed as a TracyWidom distribution:

Sample the scaled largest eigenvalue:

Check goodness of fit with TracyWidomDistribution:

Algebraically independent components of a symmetric Wishart matrix have a known PDF:

Build the distribution of independent components of a Wishart matrix:

Find the joint distribution of a diagonal element:

Use MatrixPropertyDistribution to sample diagonal elements of Wishart matrices:

Check goodness of fit:

## Properties & Relations(4)

Use MatrixPropertyDistribution to represent the scaled eigenvalues of a Wishart random matrix with identity covariance:

The limiting distribution of eigenvalues follows MarchenkoPasturDistribution:

Compare the histogram of the eigenvalues with the PDF:

The expression , where and are, respectively, an independent Gaussian vector and Wishart matrix, follows HotellingTSquareDistribution:

Use MatrixPropertyDistribution to sample expressions :

Diagonal elements of a Wishart random matrix each follow a scaled χ2 distribution:

Test against applicably scaled χ2 distributions:

Diagonal elements are not independent:

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

Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).

#### Text

Wolfram Research (2015), WishartMatrixDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html (updated 2017).

#### CMS

Wolfram Language. 2015. "WishartMatrixDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html.

#### APA

Wolfram Language. (2015). WishartMatrixDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html

#### BibTeX

@misc{reference.wolfram_2022_wishartmatrixdistribution, author="Wolfram Research", title="{WishartMatrixDistribution}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}", note=[Accessed: 24-March-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2022_wishartmatrixdistribution, organization={Wolfram Research}, title={WishartMatrixDistribution}, year={2017}, url={https://reference.wolfram.com/language/ref/WishartMatrixDistribution.html}, note=[Accessed: 24-March-2023 ]}