Maximal Invariants Over Symmetric Cones

TL;DR

This paper develops hypothesis tests for Wishart distributions over symmetric cones, deriving eigenvalue densities and proposing a generalized Bartlett's test for various algebraic structures.

Contribution

It introduces a unified framework for testing covariance structures over symmetric cones, extending classical multivariate tests to complex, quaternion, Lorentz, and octonion cases.

Findings

Derived joint eigenvalue density for generalized Wishart distributions.

Proposed a test statistic analogous to Bartlett's test for symmetric cones.

Extended covariance hypothesis testing to diverse algebraic structures.

Abstract

In this paper we consider some hypothesis tests within a family of Wishart distributions, where both the sample space and the parameter space are symmetric cones. For such testing problems, we first derive the joint density of the ordered eigenvalues of the generalized Wishart distribution and propose a test statistic analog to that of classical multivariate statistics for testing homoscedasticity of covariance matrix. In this generalization of Bartlett's test for equality of variances to hypotheses of real, complex, quaternion, Lorentz and octonion types of covariance structures.