Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices
Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices
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DOI：
10.1214/22-aos2183
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发表时间：
2020-09
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影响因子：
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通讯作者：
Zhixiang Zhang;Shu-rong Zheng;G. Pan;Pingshou Zhong
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文献类型：
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作者：
Zhixiang Zhang;Shu-rong Zheng;G. Pan;Pingshou Zhong
我们考虑一般的高维尖刺样本协方差模型，并证明当样本大小和维数成比例时，它们的首样本尖刺特征值及其线性谱统计量是渐近独立的。作为一个副产品，我们还通过消除文献中通常需要的总体协方差矩阵上的块对角假设，建立了先导样本峰值特征值的中心极限定理。此外，我们提出了尖峰总体特征向量的$L_4$范数的一致估计。基于这些结果，我们开发了一个新的统计量来检验两个尖刺总体协方差矩阵的相等性。数值研究表明，新的测试方法比现有的一些方法更有效。
We consider general high-dimensional spiked sample covariance models and show that their leading sample spiked eigenvalues and their linear spectral statistics are asymptotically independent when the sample size and dimension are proportional to each other. As a byproduct, we also establish the central limit theorem of the leading sample spiked eigenvalues by removing the block diagonal assumption on the population covariance matrix, which is commonly needed in the literature. Moreover, we propose consistent estimators of the $L_4$ norm of the spiked population eigenvectors. Based on these results, we develop a new statistic to test the equality of two spiked population covariance matrices. Numerical studies show that the new test procedure is more powerful than some existing methods.