Some sphericity tests for high dimensional data based on ratio of the traces of sample covariance matrices

Abstract

In this paper, we investigate the sphericity test for high dimensional data. The test statistic is proposed based on the ratio of traces of sample covariance matrices. The asymptotic distributions of the test statistics under the null hypothesis are established by linear spectral statistics of large sample covariance matrices. We also obtain the asymptotic power functions in the case of the spiked population model as a specific alternative. Compared with some existing tests, the proposed test can handle data having unknown means and non-Gaussian population with general fourth moment. The numerical performance demonstrates that the proposed tests have satisfactory properties in terms of size and power.