This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size. The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations. Further, some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity. Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.

中文翻译：

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基于随机矩阵理论的高维协方差矩阵的球形性和同一性检验

本文讨论了当数据维数超过样本大小时测试球形度和高维总体协方差矩阵的恒等性的问题。样本协方差矩阵特征值的前四个矩的中心极限定理是使用随机矩阵理论推导的，用于总体分布的总体。此外，在原假设下，由于数据维和样本大小都趋于无穷大，因此提供了所建议的检验统计量的一些理想渐近性质。仿真表明，所提出的测试具有比现有的加标协方差模型更大的功效。