Testing Separability of High-Dimensional Covariance Matrices

TL;DR

This paper introduces new high-dimensional invariant tests for covariance matrix separability, addressing limitations of existing methods by leveraging the core covariance matrix and enabling exact null distribution simulation.

Contribution

The paper develops novel invariant tests based on the core covariance matrix, improving high-dimensional separability testing accuracy and power.

Findings

Tests are well-defined in high-dimensional settings.

Null distributions can be exactly simulated.

Proposed tests outperform existing procedures in power.

Abstract

Due to their parsimony, separable covariance models have been popular in modeling matrix-variate data. However, the inference from such a model may be misleading if the population covariance matrix $\Sigma$ is actually non-separable, motivating the use of statistical tests of separability. The existing separability tests suffer mainly from two issues: 1) test statistics that are not well-defined in high-dimensional settings, 2) low power for small sample sizes and null distributions that depend on unknown parameters, preventing exact error rate control. To address these issues, we propose novel invariant tests using the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of $\Sigma$ is equivalent to testing sphericity of its core component. With this insight, we construct test statistics that are well-defined in high-dimensional settings and have distributions that are invariant under the null hypothesis of separability, allowing for exact simulation of null distributions. We establish the asymptotic properties of some test statistics by proving the asymptotic spectral equivalence between the sample covariance matrix and its core in a $p/n\rightarrow\gamma\in(0,\infty)$ regime. The large power of our proposed tests relative to existing procedures is demonstrated numerically.