Testing the equality of two covariance matrices is a fundamental problem in statistics, and especially challenging when the data are high-dimensional. A common approach is to obtain consistent estimates of the covariance matrices before constructing a test statistic. However, estimating the covariance matrices for high-dimensional is also difficult, albeit sometimes alleviated by assumptions such as sparsity. Through a novel use of random integration, we can test the equality of high-dimensional covariance matrices without estimating them and 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 high-dimensional factor 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 difficult settings when there are only a few large or many small diagonal disturbances between the two covariance matrices.

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