论文标题
大型样品协方差矩阵的尖峰特征值和线性光谱统计的渐近独立性
Asymptotic independence of spiked eigenvalues and linear spectral statistics for large sample covariance matrices
论文作者
论文摘要
我们考虑一般的高维峰值样品协方差模型，并表明当样本量和维度相互成比例时，它们的领先样品峰值特征值及其线性光谱统计在渐近独立。作为副产品，我们还通过消除对种群协方差矩阵的块对角假设来建立尖刺特征值的中心极限定理，这在文献中通常是必不可少的。此外，我们提出了对峰值人口特征向量的$ L_4 $规范的一致估计器。基于这些结果，我们开发了一个新的统计数据，以测试两个峰值种群协方差矩阵的平等。数值研究表明，新的测试程序比某些现有方法更强大。
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.
