Abstract: Testing the equality of two covariance matrices is a fundamental problem in statistics, and especially challenging when the data are high dimensional. By means of a novel use of random integration, we test the equality of high-dimensional covariance matrices without assuming parametric distributions for the two underlying populations, even if the dimension is much larger than the sample size. The asymptotic properties of our test for an arbitrary number of covariates and sample size are studied in depth under a general multivariate model. The finite-sample performance of our test is evaluated using numerical studies. The empirical results demonstrate that the proposed test is highly competitive with existing tests in a wide range of settings, and particularly powerful when there exist a few large or many small diagonal disturbances between the two covariance matrices.

Key words and phrases: Covariance matrix, high-dimensional data, random integration.