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

Frédéric Ouimet

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 Lukić (2024). Unlike the orthogonally invariant Hankel-transform test of Lukić and Milošević (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.

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

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 -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 Lukić (2024). Unlike the orthogonally invariant Hankel-transform test of Lukić and Milošević (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.
Paper Structure (9 sections, 7 theorems, 239 equations)

This paper contains 9 sections, 7 theorems, 239 equations.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Main results
  4. Conclusion and outlook
  5. Proofs
  6. Proof of Proposition \ref{['prop:test.statistic']}
  7. Proof of Theorem \ref{['thm:null.shrinking.bandwidth']}
  8. Proof of Theorem \ref{['thm:null.fixed.bandwidth']}
  9. Technical lemmas

Key Result

Proposition 3.1

For $i,\ell\in \{1,2\}$, let $\delta_{i,\ell}$ denote the usual Kronecker delta, i.e., $\delta_{i,\ell}=1$ or $0$ according to whether $i = \ell$ or $i \neq \ell$, and let $a_{i,\ell} = \delta_{i,\ell} - (n_{\ell}/n)$. We have

Theorems & Definitions (11)

  • Proposition 3.1
  • Theorem 3.1: Null limit in the common shrinking-bandwidth regime
  • Theorem 3.2: Fixed-bandwidth null limit as a weighted chi-square series
  • Lemma B.1
  • proof : Proof of Lemma \ref{['lem:kappa.diagonal.p.moment']}
  • Lemma B.2
  • proof : Proof of Lemma \ref{['lem:kappa.offdiag.p.moment']}
  • Lemma B.3
  • proof : Proof of Lemma \ref{['lem:h.centered.p.moment.control']}
  • Lemma B.4
  • ...and 1 more