Testing high-dimensional covariance matrices with random projections and corrected likelihood ratio

Testing the equality between two high-dimensional covariance matrices is challenging. As the efficient way to measure evidential discrepancy from observed data, the likelihood ratio test is expected to be powerful when the null hypothesis is violated. However, when the data dimensionality becomes large and may substantially exceed the sample size, likelihood ratio based approaches are encountering both practical and theoretical difficulties. To solve the problem, we propose in this study to first randomly project the original high-dimensional data to some lower-dimensional space, and then to apply the corrected likelihood ratio tests developed with the random matrix theory. We show that our test is consistent under the null hypothesis. Through evaluating the power function which is a challenging objective in this context, we show evidence that our test based on random projection matrix with reasonable column size is more powerful when the two covariance matrices are unequal but component-wise discrepancy could be small - a weak and dense signal setting. Numerical studies with simulations and a real data analysis confirm the merits of our test.