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.

测试协方差矩阵的球形度已成为当代统计中的活跃部分。本文基于似然比检验（LRT），提出了一种新的高维球形度检验方法。提出的测试扩大了LRT的适用性，当尺寸大于样本大小时，LRT失败。在具有有限第四矩的一般总体下，检验统计量在原假设下显示为渐近正态分布。当替代假设为真时，在加标模型下得出检验统计量的极限分布。仿真研究表明，所提出的测试可以很好地控制I型错误率，并且在几种检查情况下的经验能力方面都优于某些知名测试。