High-dimensional sphericity test by extended likelihood ratio

Abstract

Testing sphericity of the covariance matrices has been an active part in contemporary statistics. In this paper, we put forward a new test procedure for high-dimensional sphericity test based on the likelihood ratio test (LRT). The proposed test broadens the applicability of LRT which fails when the dimension is larger than the sample size. Under general population with finite fourth moment, the test statistic is shown to be asymptotically normally distributed under the null hypothesis. When the alternative hypothesis is true, the limiting distribution of the test statistic is derived under the spiked model. Simulation studies reveal that the proposed test controls the Type I error rate very well and outperforms some well-known tests in terms of the empirical power in several examined situations.