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Asymptotic testing of covariance separability for matrix elliptical data

Joni Virta, Takeru Matsuda

TL;DR

This work develops an asymptotic framework for testing covariance separability of matrix-valued data under a matrix-elliptical model that encompasses Gaussian and matrix t distributions. It introduces a determinant-scaled, Frobenius-norm based test t_n and a Wald-type counterpart w_n, deriving their null distributions as a chi-square mixture whose weights depend on the fourth-moment structure of the underlying matrix Z. The analysis combines Fisher-consistent estimators for separable and non-separable covariance structures, explicit asymptotic linearizations, and a detailed fourth-moment theory for matrix spherical distributions, enabling fast, permutation-free testing across broad distributional families. Simulations demonstrate robustness to heavy tails and power competitive with Gaussian LRT under normal data, highlighting practical benefits for high-dimensional matrix-elliptical settings. The results provide a flexible, fast, and distributionally robust approach to covariance separability testing with clear guidance for estimating the necessary moment weights from data.

Abstract

We propose a new asymptotic test for the separability of a covariance matrix. The null distribution is valid in wide matrix elliptical model that includes, in particular, both matrix Gaussian and matrix $t$-distribution. The test is fast to compute and makes no assumptions about the component covariance matrices. An alternative, Wald-type version of the test is also proposed. Our simulations reveal that both versions of the test have good power even for heavier-tailed distributions and can compete with the Gaussian likelihood ratio test in the case of normal data.

Asymptotic testing of covariance separability for matrix elliptical data

TL;DR

This work develops an asymptotic framework for testing covariance separability of matrix-valued data under a matrix-elliptical model that encompasses Gaussian and matrix t distributions. It introduces a determinant-scaled, Frobenius-norm based test t_n and a Wald-type counterpart w_n, deriving their null distributions as a chi-square mixture whose weights depend on the fourth-moment structure of the underlying matrix Z. The analysis combines Fisher-consistent estimators for separable and non-separable covariance structures, explicit asymptotic linearizations, and a detailed fourth-moment theory for matrix spherical distributions, enabling fast, permutation-free testing across broad distributional families. Simulations demonstrate robustness to heavy tails and power competitive with Gaussian LRT under normal data, highlighting practical benefits for high-dimensional matrix-elliptical settings. The results provide a flexible, fast, and distributionally robust approach to covariance separability testing with clear guidance for estimating the necessary moment weights from data.

Abstract

We propose a new asymptotic test for the separability of a covariance matrix. The null distribution is valid in wide matrix elliptical model that includes, in particular, both matrix Gaussian and matrix -distribution. The test is fast to compute and makes no assumptions about the component covariance matrices. An alternative, Wald-type version of the test is also proposed. Our simulations reveal that both versions of the test have good power even for heavier-tailed distributions and can compete with the Gaussian likelihood ratio test in the case of normal data.
Paper Structure (13 sections, 21 theorems, 108 equations, 2 figures)

This paper contains 13 sections, 21 theorems, 108 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Estimation theory
  3. Population level
  4. Sample level
  5. Fourth moments of spherical matrices
  6. Hypothesis tests for covariance separability
  7. Test based on $n \| V_n - I_{p_1 p_2} \|_F^2$
  8. Wald-type version of the test
  9. Simulations
  10. Proofs of technical results
  11. Proof for Section \ref{['sec:estimation']}
  12. Proof for Section \ref{['sec:fourth_moment_Z']}
  13. Proof for Section \ref{['sec:h0_test']}

Key Result

Theorem 1

Under $P(M, \Sigma_1, \Sigma_2, F)$, the limiting distribution of $n \| V_n - I_{p_1 p_2} \|_F^2$, as $n \rightarrow \infty$, is where the two $\chi^2$-variates are independent.

Figures (2)

  • Figure 1: Average rejection rates (over 2000 replicates) of our proposed test and the Gaussian LRT for varying matrix $t$-distributions, sample sizes $n$ and local deviations $\tau$ from the null hypothesis when $(p_1, p_2) = (3, 3)$.
  • Figure 2: Average rejection rates (over 2000 replicates) of our proposed test and the Gaussian LRT for varying matrix $t$-distributions, sample sizes $n$ and local deviations $\tau$ from the null hypothesis when $(p_1, p_2) = (5, 5)$.

Theorems & Definitions (39)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Lemma 1
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Lemma 2
  • ...and 29 more