## Sample Covariance Matrix

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### Zhidong Bai - One of the best experts on this subject based on the ideXlab platform.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
Journal of Statistical Planning and Inference, 2013
Co-Authors: Jiaqi Chen, Zhidong Bai, Yingli Qin, Jianfeng Yao
Abstract:

Abstract This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko–Pastur equation, originally defined in the complex plane, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P 500 daily stock returns is also given.

• ##### Convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
The Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mar\v{c}enko-Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to Sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.

• ##### convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mary cenko– Pastur distribution function is of order O(N −1/2 ). Given that data dimension n to Sample size N ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed η> 0, the convergence rates of VESD are O(N −1/4 ) in probability and O(N −1/4+η ) almost surely, requiring finite 8th moment of the underlying distribution.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
arXiv: Methodology, 2013
Co-Authors: Jiaqi Chen, Jianfeng Yao, Yingli Qin, Zhidong Bai
Abstract:

This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko-Pastur equation, originally defined in the complex plan, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P stocks data is also given.

• ##### asymptotic properties of eigenmatrices of a large Sample Covariance Matrix
arXiv: Probability, 2011
Co-Authors: Zhidong Bai, H X Liu, Wingkeung Wong
Abstract:

Let $S_n=\frac{1}{n}X_nX_n^*$ where $X_n=\{X_{ij}\}$ is a $p\times n$ Matrix with i.i.d. complex standardized entries having finite fourth moments. Let $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)=\sqrt{p}({\mathbf {x}}_n(\mathbf {t}_1)^*(S_n+\sigma I)^{-1}{\mathbf {x}}_n(\mathbf {t}_2)-{\mathbf {x}}_n(\mathbf {t}_1)^*{\mathbf {x}}_n(\mathbf {t}_2)m_n(\sigma))$ in which $\sigma>0$ and $m_n(\sigma)=\int\frac{dF_{y_n}(x)}{x+\sigma}$ where $F_{y_n}(x)$ is the Marcenko--Pastur law with parameter $y_n=p/n$; which converges to a positive constant as $n\to\infty$, and ${\mathbf {x}}_n(\mathbf {t}_1)$ and ${\mathbf {x}}_n(\mathbf {t}_2)$ are unit vectors in ${\Bbb{C}}^p$, having indices $\mathbf {t}_1$ and $\mathbf {t}_2$, ranging in a compact subset of a finite-dimensional Euclidean space. In this paper, we prove that the sequence $Y_n(\mathbf {t}_1,\mathbf {t}_2,\sigma)$ converges weakly to a $(2m+1)$-dimensional Gaussian process. This result provides further evidence in support of the conjecture that the distribution of the eigenMatrix of $S_n$ is asymptotically close to that of a Haar-distributed unitary Matrix.

### Yingli Qin - One of the best experts on this subject based on the ideXlab platform.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
Journal of Statistical Planning and Inference, 2013
Co-Authors: Jiaqi Chen, Zhidong Bai, Yingli Qin, Jianfeng Yao
Abstract:

Abstract This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko–Pastur equation, originally defined in the complex plane, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P 500 daily stock returns is also given.

• ##### Convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
The Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mar\v{c}enko-Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to Sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.

• ##### convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mary cenko– Pastur distribution function is of order O(N −1/2 ). Given that data dimension n to Sample size N ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed η> 0, the convergence rates of VESD are O(N −1/4 ) in probability and O(N −1/4+η ) almost surely, requiring finite 8th moment of the underlying distribution.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
arXiv: Methodology, 2013
Co-Authors: Jiaqi Chen, Jianfeng Yao, Yingli Qin, Zhidong Bai
Abstract:

This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko-Pastur equation, originally defined in the complex plan, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P stocks data is also given.

### Thomas Mikosch - One of the best experts on this subject based on the ideXlab platform.

• ##### the eigenvalues of the Sample Covariance Matrix of a multivariate heavy tailed stochastic volatility model
Bernoulli, 2018
Co-Authors: Anja Janßen, Thomas Mikosch, Mohsen Rezapour, Xiaolei Xie
Abstract:

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-Sample behavior of its Sample Covariance Matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the Sample Covariance Matrix. For the case of heavy-tailed innovations, we show that the limiting behavior resembles that of completely independent observations. In contrast to this, for a heavy-tailed volatility sequence the possible limiting behavior is more diverse and allows for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

• ##### eigenvalues and eigenvectors of heavy tailed Sample Covariance matrices with general growth rates the iid case
Stochastic Processes and their Applications, 2017
Co-Authors: Johannes Heiny, Thomas Mikosch
Abstract:

Abstract In this paper we study the joint distributional convergence of the largest eigenvalues of the Sample Covariance Matrix of a p -dimensional time series with iid entries when p converges to infinity together with the Sample size n . We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov (2004, 2006) and Auffinger et al. (2009) proved the weak convergence of the point processes of the normalized eigenvalues of the Sample Covariance Matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Frechet distributed random variable. They proved these results under the assumption that p and n are proportional to each other. In this paper we show that the aforementioned results remain valid if p grows at any polynomial rate. The proofs are different from those in Auffinger et al. (2009) and Soshnikov (2004, 2006); we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the Sample Covariance Matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to Sample autoCovariance matrices.

• ##### eigenvalues and eigenvectors of heavy tailed Sample Covariance matrices with general growth rates the iid case
arXiv: Probability, 2016
Co-Authors: Johannes Heiny, Thomas Mikosch
Abstract:

In this paper we study the joint distributional convergence of the largest eigenvalues of the Sample Covariance Matrix of a $p$-dimensional time series with iid entries when $p$ converges to infinity together with the Sample size $n$. We consider only heavy-tailed time series in the sense that the entries satisfy some regular variation condition which ensures that their fourth moment is infinite. In this case, Soshnikov [31, 32] and Auffinger et al. [2] proved the weak convergence of the point processes of the normalized eigenvalues of the Sample Covariance Matrix towards an inhomogeneous Poisson process which implies in turn that the largest eigenvalue converges in distribution to a Frechet distributed random variable. They proved these results under the assumption that $p$ and $n$ are proportional to each other. In this paper we show that the aforementioned results remain valid if $p$ grows at any polynomial rate. The proofs are different from those in [2, 31, 32]; we employ large deviation techniques to achieve them. The proofs reveal that only the diagonal of the Sample Covariance Matrix is relevant for the asymptotic behavior of the largest eigenvalues and the corresponding eigenvectors which are close to the canonical basis vectors. We also discuss extensions of the results to Sample autoCovariance matrices.

• ##### The eigenvalues of the Sample Covariance Matrix of a multivariate heavy-tailed stochastic volatility model
arXiv: Probability, 2016
Co-Authors: Anja Janßen, Thomas Mikosch, Mohsen Rezapour, Xiaolei Xie
Abstract:

We consider a multivariate heavy-tailed stochastic volatility model and analyze the large-Sample behavior of its Sample Covariance Matrix. We study the limiting behavior of its entries in the infinite-variance case and derive results for the ordered eigenvalues and corresponding eigenvectors. Essentially, we consider two different cases where the tail behavior either stems from the i.i.d. innovations of the process or from its volatility sequence. In both cases, we make use of a large deviations technique for regularly varying time series to derive multivariate $\alpha$-stable limit distributions of the Sample Covariance Matrix. While we show that in the case of heavy-tailed innovations the limiting behavior resembles that of completely independent observations, we also derive that in the case of a heavy-tailed volatility sequence the possible limiting behavior is more diverse, i.e. allowing for dependencies in the limiting distributions which are determined by the structure of the underlying volatility sequence.

• ##### asymptotic theory for the Sample Covariance Matrix of a heavy tailed multivariate time series
Stochastic Processes and their Applications, 2016
Co-Authors: Richard A. Davis, Thomas Mikosch, Oliver Pfaffel
Abstract:

Abstract In this paper we give an asymptotic theory for the eigenvalues of the Sample Covariance Matrix of a multivariate time series. The time series constitutes a linear process across time and between components. The input noise of the linear process has regularly varying tails with index α ∈ ( 0 , 4 ) ; in particular, the time series has infinite fourth moment. We derive the limiting behavior for the largest eigenvalues of the Sample Covariance Matrix and show point process convergence of the normalized eigenvalues. The limiting process has an explicit form involving points of a Poisson process and eigenvalues of a non-negative definite Matrix. Based on this convergence we derive limit theory for a host of other continuous functionals of the eigenvalues, including the joint convergence of the largest eigenvalues, the joint convergence of the largest eigenvalue and the trace of the Sample Covariance Matrix, and the ratio of the largest eigenvalue to their sum.

### Ningning Xia - One of the best experts on this subject based on the ideXlab platform.

• ##### Convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
The Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mar\v{c}enko-Pastur distribution function is of order $O(N^{-1/2})$. Given that data dimension $n$ to Sample size $N$ ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed $\eta>0$, the convergence rates of VESD are $O(N^{-1/4})$ in probability and $O(N^{-1/4+\eta})$ almost surely, requiring finite 8th moment of the underlying distribution.

• ##### convergence rates of eigenvector empirical spectral distribution of large dimensional Sample Covariance Matrix
Annals of Statistics, 2013
Co-Authors: Ningning Xia, Yingli Qin, Zhidong Bai
Abstract:

The eigenvector Empirical Spectral Distribution (VESD) is adopted to investigate the limiting behavior of eigenvectors and eigenvalues of Covariance matrices. In this paper, we shall show that the Kolmogorov distance between the expected VESD of Sample Covariance Matrix and the Mary cenko– Pastur distribution function is of order O(N −1/2 ). Given that data dimension n to Sample size N ratio is bounded between 0 and 1, this convergence rate is established under finite 10th moment condition of the underlying distribution. It is also shown that, for any fixed η> 0, the convergence rates of VESD are O(N −1/4 ) in probability and O(N −1/4+η ) almost surely, requiring finite 8th moment of the underlying distribution.

### Jianfeng Yao - One of the best experts on this subject based on the ideXlab platform.

• ##### Asymptotic joint distribution of extreme eigenvalues and trace of large Sample Covariance Matrix in a generalized spiked population model
arXiv: Statistics Theory, 2019
Co-Authors: Fang Han, Jianfeng Yao
Abstract:

This paper studies the joint limiting behavior of extreme eigenvalues and trace of large Sample Covariance Matrix in a generalized spiked population model, where the asymptotic regime is such that the dimension and Sample size grow proportionally. The form of the joint limiting distribution is applied to conduct Johnson-Graybill-type tests, a family of approaches testing for signals in a statistical model. For this, higher order correction is further made, helping alleviate the impact of finite-Sample bias. The proof rests on determining the joint asymptotic behavior of two classes of spectral processes, corresponding to the extreme and linear spectral statistics respectively.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
Journal of Statistical Planning and Inference, 2013
Co-Authors: Jiaqi Chen, Zhidong Bai, Yingli Qin, Jianfeng Yao
Abstract:

Abstract This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko–Pastur equation, originally defined in the complex plane, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P 500 daily stock returns is also given.

• ##### estimation of the population spectral distribution from a large dimensional Sample Covariance Matrix
arXiv: Methodology, 2013
Co-Authors: Jiaqi Chen, Jianfeng Yao, Yingli Qin, Zhidong Bai
Abstract:

This paper introduces a new method to estimate the spectral distribution of a population Covariance Matrix from high-dimensional data. The method is founded on a meaningful generalization of the seminal Marcenko-Pastur equation, originally defined in the complex plan, to the real line. Beyond its easy implementation and the established asymptotic consistency, the new estimator outperforms two existing estimators from the literature in almost all the situations tested in a simulation experiment. An application to the analysis of the correlation Matrix of S&P stocks data is also given.

• ##### fluctuations of an improved population eigenvalue estimator in Sample Covariance Matrix models
arXiv: Probability, 2011
Co-Authors: Jianfeng Yao, Jamal Najim, Romain Couillet, Merouane Debbah
Abstract:

This article provides a central limit theorem for a consistent estimator of population eigenvalues with large multiplicities based on Sample Covariance matrices. The focus is on limited Sample size situations, whereby the number of available observations is known and comparable in magnitude to the observation dimension. An exact expression as well as an empirical, asymptotically accurate, approximation of the limiting variance is derived. Simulations are performed that corroborate the theoretical claims. A specific application to wireless sensor networks is developed.

• ##### ON ESTIMATION OF THE POPULATION SPECTRAL DISTRIBUTION FROM A HIGH-DIMENSIONAL Sample Covariance Matrix
Australian & New Zealand Journal of Statistics, 2010
Co-Authors: Zhidong Bai, Jiaqi Chen, Jianfeng Yao
Abstract:

Sample Covariance matrices play a central role in numerous popular statistical methodologies, for example principal components analysis, Kalman filtering and independent component analysis. However, modern random Matrix theory indicates that, when the dimension of a random vector is not negligible with respect to the Sample size, the Sample Covariance Matrix demonstrates significant deviations from the underlying population Covariance Matrix. There is an urgent need to develop new estimation tools in such cases with high-dimensional data to recover the characteristics of the population Covariance Matrix from the observed Sample Covariance Matrix. We propose a novel solution to this problem based on the method of moments. When the parametric dimension of the population spectrum is finite and known, we prove that the proposed estimator is strongly consistent and asymptotically Gaussian. Otherwise, we combine the first estimation method with a cross-validation procedure to select the unknown model dimension. Simulation experiments demonstrate the consistency of the proposed procedure. We also indicate possible extensions of the proposed estimator to the case where the population spectrum has a density.