# MatrixTDistribution

MatrixTDistribution[Σrow,Σcol,ν]

represents zero mean matrix distribution with row covariance matrix Σrow, column covariance matrix Σcol, and degrees of freedom parameter ν.

MatrixTDistribution[μ,Σrow,Σcol,ν]

represents matrix distribution with mean matrix μ.

# Details • The probability density for a matrix of dimensions in a matrix distribution is proportional to with an identity matrix of length .
• MatrixTDistribution[Σrow,Σcol,ν] is the distribution of MatrixNormalDistribution[Σ,Σcol] with sampled from InverseWishartMatrixDistribution[ν+n-1,Σrow].
• MatrixTDistribution[μ,c Σrow,c-1 Σcol,ν] has the same distribution as MatrixTDistribution[μ,Σrow,Σcol,ν] for any positive real constant c.
• The covariance matrices Σrow and Σcol can be any symmetric positive definite matrices of real numbers of dimensions {n,n} and {m,m}, respectively. The degrees of freedom parameter ν can be any positive number, and the mean matrix μ can be any matrix of real numbers of dimensions {n,m}.
• MatrixTDistribution can be used with such functions as MatrixPropertyDistribution, EstimatedDistribution, and RandomVariate.

# Examples

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## Basic Examples(2)

Sample from matrix distribution:

Mean and variance:

## Scope(6)

Generate a single pseudorandom matrix:

Generate a single pseudorandom matrix with nonzero mean:

Generate a set of pseudorandom matrices:

Sample at extended precision:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare LogLikelihood for both distributions:

Skewness and kurtosis:

Probability density function:

Plot PDF for a diagonal matrices:

## Properties & Relations(4)

Matrix t distribution is defined up to a positive multiplicative constant:

Equivalent distribution with row and column scale matrices multiplied and divided by a positive constant:

Compute the PDF of the distributions at a random point:

MatrixTDistribution[Σrow,Σcol,ν] is a parameter mixture of MatrixNormalDistribution[Σ,Σcol] with following InverseWishartMatrixDistribution[ν+n-1,Σrow]:

Create a sample following the parameter mixture of MatrixNormalDistribution with InverseWishartMatrixDistribution:

Fit the sample data to MatrixTDistribution:

Compute log-likelihood ratio statistic against the appropriate MatrixTDistribution

Log-likelihood ratio follows ChiSquareDistribution with the parameter equal to the number of degrees of freedom:

Compute the -value of log-likelihood ratio test:

For matrix sampled from matrix distribution, the expression follows Student distribution for any nonzero vectors and with lengths that match with the dimension of :

Use MatrixPropertyDistribution to sample values of the expression :

Check agreement with the expected distribution:

For matrix sampled from matrix distribution, follows multivariate distribution for any nonzero vector with length that matches with the number of columns of :

Use MatrixPropertyDistribution to sample values of :

Verify goodness of fit with the expected distribution:

## Possible Issues(1)

Matrix distribution is defined up to a multiplicative scaling constant. The estimated parameters may not be close to the ones that specify the underlying distribution:

Sample from the matrix distribution:

Estimate the distribution:

Compare the estimated scale parameters with the ones of the underlying distribution:

Kronecker products of the scale matrices are close to each other:

The LogLikelihood of the distributions indicate that the estimate is good: