Equality tests of covariance matrices under a low-dimensional factor structure

We propose an equality test to compare two covariance matrices in a high-dimensional framework while accommodating a low-dimensional latent factor model. We show that null limiting distributions of the test statistics follow a weighted mixture of chi-square distributions under a high-dimensional asymptotic regime combined with weak technical conditions. This distribution depends on the noise covariance matrix and the number of latent factors. Because latent factors are often unknown, we employ an estimation that builds on recent advances in random matrix theory. A numerical study demonstrates the asymptotic power of the proposed test and confirms its favorable analytical properties compared to existing procedures.