## The Annals of Statistics

- Ann. Statist.
- Volume 43, Number 2 (2015), 546-591.

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

Shurong Zheng, Zhidong Bai, and Jianfeng Yao

#### Abstract

Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of high-dimensional noncentralized 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 noncentralized 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 CLTs for noncentralized sample covariance matrices when the Gaussian-like moment conditions do not hold but also characterize the nonnegligible differences among the CLTs 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 CLTs, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLTs for the ME and unbiased sample covariance matrix is nonnegligible 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.

#### Article information

**Source**

Ann. Statist., Volume 43, Number 2 (2015), 546-591.

**Dates**

First available in Project Euclid: 24 February 2015

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1424787428

**Digital Object Identifier**

doi:10.1214/14-AOS1292

**Mathematical Reviews number (MathSciNet)**

MR3316190

**Zentralblatt MATH identifier**

1312.62074

**Subjects**

Primary: 62H15: Hypothesis testing 15B52: Random matrices 62H10: Distribution of statistics

**Keywords**

CLT for linear spectral statistics unbiased sample covariance matrix substitution principle testing on high-dimensional covariance matrix high-dimensional sample covariance matrix large Fisher matrix high-dimensional data

#### Citation

Zheng, Shurong; Bai, Zhidong; Yao, Jianfeng. Substitution principle for CLT of linear spectral statistics of high-dimensional sample covariance matrices with applications to hypothesis testing. Ann. Statist. 43 (2015), no. 2, 546--591. doi:10.1214/14-AOS1292. https://projecteuclid.org/euclid.aos/1424787428