Hypothesis testing for high-dimensional covariance matrices

This paper discusses the problem of testing for high-dimensional covariance matrices. Tests for an identity matrix and for the equality of two covariance matrices are considered when the data dimension and the sample size are both large. Most importantly, the dimension can be much larger than the sample size. The proposed test statistics are built upon the Stieltjes transform of the spectral distribution of the sample covariance matrix. We prove that the proposed statistics are asymptotically chi-square distributed under the null hypotheses, and normally distributed under the alternative hypotheses. Simulation results show that for finite dimension and sample size the proposed tests outperform some existing methods in various cases.