We consider hypothesis testing for high-dimensional covariance structures in which the covariance matrix is a (i) scaled identity matrix, (ii) diagonal matrix, or (iii) intraclass covariance matrix. Our purpose is to systematically establish a nonparametric approach for testing the high-dimensional covariance structures (i)–(iii). We produce a new common test statistic for each covariance structure and show that the test statistic is an unbiased estimator of its corresponding test parameter. We prove that the test statistic establishes the asymptotic normality. We propose a new test procedure for (i)–(iii) and evaluate its asymptotic size and power theoretically when both the dimension and sample size increase. We investigate the performance of the proposed test procedure in simulations. As an application of testing the covariance structures, we give a test procedure to identify an eigenvector. Finally, we demonstrate the proposed test procedure by using a microarray data set.

中文翻译：

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高维协方差结构的假设检验

我们考虑对高维协方差结构进行假设检验，其中协方差矩阵是 (i) 缩放单位矩阵、(ii) 对角矩阵或 (iii) 类内协方差矩阵。我们的目的是系统地建立一种非参数方法来测试高维协方差结构（i）-（iii）。我们为每个协方差结构生成一个新的通用检验统计量，并表明检验统计量是其相应检验参数的无偏估计量。我们证明检验统计量建立了渐近正态性。我们为 (i)-(iii) 提出了一种新的测试程序，并在维度和样本大小都增加时从理论上评估其渐近大小和功效。我们研究了所提出的测试程序在模拟中的性能。作为测试协方差结构的应用，我们给出了一个测试程序来识别特征向量。最后，我们通过使用微阵列数据集演示了建议的测试程序。