Abstract In this article, we consider a test of the sphericity for high-dimensional covariance matrices. We produce a test statistic by using the extended cross-data-matrix (ECDM) methodology. We show that the ECDM test statistic is based on an unbiased estimator of a sphericity measure. In addition, the ECDM test statistic enjoys consistency properties and the asymptotic normality in high-dimensional settings. We propose a new test procedure based on the ECDM test statistic and evaluate its asymptotic size and power theoretically and numerically. We give a two-stage sampling scheme so that the test procedure can ensure a prespecified level both for the size and power. We apply the test procedure to detect divergently spiked noise in high-dimensional statistical analysis. We analyze gene expression data by the proposed test procedure.

中文翻译：

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高维数据球形度检验及其在发散尖峰噪声检测中的应用

摘要 在本文中，我们考虑了高维协方差矩阵的球度测试。我们使用扩展的交叉数据矩阵 (ECDM) 方法生成测试统计量。我们表明 ECDM 检验统计量基于球形度量的无偏估计量。此外，ECDM 检验统计量在高维设置中具有一致性属性和渐近正态性。我们提出了一种基于 ECDM 测试统计量的新测试程序，并从理论上和数值上评估其渐近大小和功效。我们给出了一个两阶段抽样方案，以便测试程序可以确保大小和功率都达到预先指定的水平。我们应用测试程序来检测高维统计分析中的发散尖峰噪声。我们通过建议的测试程序分析基因表达数据。