20: Testing Separability of High-Dimensional Covariance Matrices

Music City Center 

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 is actually not separable. This suggests the use of statistical tests of covariance separability. Likelihood ratio tests have tractable null distributions and good power when the sample size $n$ is not less than the number of variables $p$, but are not well-defined otherwise. Other existing separability tests for the $p>n$ case have low power for small sample sizes, and have null distributions that depend on unknown parameters, preventing exact error rate control. To address these issues, we propose novel invariant tests leveraging the core covariance matrix, a complementary notion to a separable covariance matrix. We show that testing separability of a covariance matrix is equivalent to testing sphericity of its core component. Based on this observation, 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 study asymptotic null distributions and show consistency of our tests in a $p/n\rightarrow(0,\infty)$ asymptotic regime. Via simulation studies, we illustrate the large power of our proposed tests as compared to existing procedures.

Core covariance matrix

eigenvalues

hypothesis testing

invariance

separable covariance matrix

separable covariance expansion 

Section on Nonparametric Statistics