Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing

By Shu-rong Zheng, Z. Bai, Jianfeng Yao. 2014

97 citations recorded by the source corpus.

Abstract. Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLT’s) for linear spectral statistics of high-dimensional non-centralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for non-centralized sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the ME (moment estimator, constructed by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator n as N = n − 1 in the ME) without depending on unknown population mean vectors. In this paper, we not only establish the new CLT’s for non-centralized sample covariance matrices when the Gaussian-like moment conditions do not hold but also characterize the non-negligible differences among the CLT’s for the three classes of high-dimensional sample covariance matrices by establishing a substitution principle: by substituting the adjusted sample size N = n − 1 for the actual sample size n in the centering term of the new CLT’s, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLT’s for the ME and unbiased sample covariance matrix is non-negligible in the centering term although the only difference between two sample covariance matrices is a normalization by n and n− 1, respectively. The new results are applied to two testing problems for high-dimensional covariance matrices.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/137f788b913d4dcf50a1ebedbfd6b5d937b4e343

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In Applied Mathematics

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Citations in the corpus

Global Testing and Large-Scale Multiple Testing for High-Dimensional Covariance Structures (2017, unknown): “Zheng, S., Z. Bai, and J. Yao (2015).”
Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation (2016, methodology): “Recently, Zheng et al. (2015) extended it to the setting with unknown population mean by applying a substitution principle established for the Central Limit Theorem for linear spectral statistics of sample covariance matrices with unknown mean.”