Extended Estimation of Matrix Distributions

Version 11 introduced random matrices, which had been tightly integrated with the existing probability and statistics framework. Random matrices have uses in a surprising variety of fields, including statistics, physics, pure mathematics, biology and finance, among others. Version 12 completes the support for random matrices with estimation for MatrixNormalDistribution, MatrixTDistribution, WishartMatrixDistribution and InverseWishartMatrixDistribution.

WishartMatrixDistribution[ν, Σ] is the distribution of the sample covariance from independent realizations of a multivariate Gaussian distribution with covariance matrix when the degrees of freedom parameter is an integer.

Simulate m random samples of length n from a MultinormalDistribution.

Compute sample covariance for each list.

The result is a list of n matrices.

Fit WishartMatrixDistribution to the covariance sample.

Compare the mean of the fitted distribution with the mean of the sample covariances.

Compare the variances.

For a matrix distributed as WishartMatrixDistribution[ν, Σ], the inverse is distributed as InverseWishartMatrixDistribution[ν, Σ^-1].

Compute the inverse of the sample covariances and fit an InverseWishartMatrixDistribution.

Check if the covariance matrix of the estimated Wishart matrix distribution is the inverse of the inverse Wishart model.