Use of random integration to test equality of high dimensional covariance matrices

Testing the equality of two covariance matrices is a fundamental problem in statistics, and especially challenging when the data are high-dimensional. Through a novel use of random integration, we can 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 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 through numerical studies. The empirical results demonstrate that our test is highly competitive with existing tests in a wide range of settings. In particular, our proposed test is distinctly powerful under different settings when there exist a few large or many small diagonal disturbances between the two covariance matrices.