Real gamma distribution on analytic bundles of flag varieties

TL;DR

This paper develops new matrix normal distributions on analytic bundles of flag varieties, generalizing covariance structures and deriving related distributions like Wishart and quadratic forms.

Contribution

It introduces four novel matrix normal distributions on flag varieties, extending existing covariance models and connecting to classical distributions as corollaries.

Findings

Derived joint distribution of sample variances and covariances.

Extended classical distributions such as Wishart and quadratic forms.

Provided new theoretical insights into covariance structures on flag varieties.

Abstract

This paper introduces four matrix normal distributions on analytic bundles of flag varieties, extending the separable covariance $\varPhi \otimes \varPsi$ with potentially variable-level ($\varPsi$) and/or sample-level ($\varPhi$) correlations. The joint distribution of sample variances and covariances, leading to the product moment distribution, is considered when precision matrices admit a specific form. Several well-known consequences, including the non-central Wishart distribution and normal quadratic forms, now appear as corollaries.