A two-sample test for symmetric positive definite matrix distributions using Wishart kernel density estimators

TL;DR

This paper introduces a nonparametric two-sample test for distributions on symmetric positive definite matrices using Wishart kernel density estimators, capable of detecting differences in eigenvector structure.

Contribution

It develops a new kernel-based two-sample test leveraging Wishart KDEs with boundary bias correction, providing explicit formulas and asymptotic distributions for the test statistic.

Findings

The test effectively detects differences in eigenvector orientations.

The method avoids numerical integration by using closed-form overlap of kernels.

It offers a competitive alternative to existing Laplace-transform based tests.

Abstract

We develop a nonparametric two-sample test for distributions supported on the cone of symmetric positive definite matrices. The procedure relies on the Wishart kernel density estimator (KDE) introduced by Belzile et al. (2025), whose support-adaptive kernel alleviates boundary bias by remaining confined to the cone. Our test statistic is the rescaled integrated squared difference between two Wishart KDEs and can be expressed as a two-sample $V$-statistic via an explicit closed-form overlap of Wishart kernels, avoiding numerical integration. Under the null hypothesis of equal densities, we derive the asymptotic distribution in both the common shrinking-bandwidth and fixed-bandwidth regimes. The proposed method provides a kernel-based competitor to the empirical Laplace-transform two-sample test of Luki\'c (2024). Unlike the orthogonally invariant Hankel-transform test of Luki\'c and Milo\v{s}evi\'c (2024), our statistic can detect alternatives that differ only through eigenvector structure, for instance, Wishart models with the same shape parameter and the same scale eigenvalues but different orientations.