scholarly journals ON DETERMINING THE NUMBER OF SPIKES IN A HIGH-DIMENSIONAL SPIKED POPULATION MODEL

2012 ◽  
Vol 01 (01) ◽  
pp. 1150002 ◽  
Author(s):  
DAMIEN PASSEMIER ◽  
JIAN-FENG YAO
Keyword(s):  
Covariance Matrix ◽  
Population Model ◽  
Factor Model ◽  
Fundamental Problem ◽  
High Dimensional ◽  
Sample Covariance ◽  
Or Economics ◽  
Linear Mixture Model ◽  
The Difference ◽  
Scientific Fields

In a spiked population model, the population covariance matrix has all its eigenvalues equal to units except for a few fixed eigenvalues (spikes). Determining the number of spikes is a fundamental problem which appears in many scientific fields, including signal processing (linear mixture model) or economics (factor model). Several recent papers studied the asymptotic behavior of the eigenvalues of the sample covariance matrix (sample eigenvalues) when the dimension of the observations and the sample size both grow to infinity so that their ratio converges to a positive constant. Using these results, we propose a new estimator based on the difference between two consecutive sample eigenvalues.

scholarly journals High dimensional covariance matrix estimation using a factor model

2008 ◽  
Vol 147 (1) ◽  
pp. 186-197 ◽  
Author(s):  
Jianqing Fan ◽  
Yingying Fan ◽  
Jinchi Lv
Keyword(s):  
Covariance Matrix ◽  
Factor Model ◽  
Covariance Matrix Estimation ◽  
High Dimensional ◽  
Matrix Estimation

scholarly journals High-Dimensional Conditional Covariance Matrices Estimation Using a Factor-GARCH Model

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 158
Author(s):  
Xiaoling Li ◽  
Xingfa Zhang ◽  
Yuan Li
Keyword(s):  
Covariance Matrix ◽  
Factor Model ◽  
Asymptotic Properties ◽  
Garch Model ◽  
Likelihood Estimation ◽  
Least Square ◽  
High Dimensional ◽  
Conditional Covariance ◽  
Financial Application ◽  
Quasi Maximum Likelihood

Estimation of a conditional covariance matrix is an interesting and important research topic in statistics and econometrics. However, modelling ultra-high dimensional dynamic (conditional) covariance structures is known to suffer from the curse of dimensionality or the problem of singularity. To partially solve this problem, this paper establishes a model by combining the ideas of a factor model and a symmetric GARCH model to describe the dynamics of a high-dimensional conditional covariance matrix. Quasi maximum likelihood estimation (QMLE) and least square estimation (LSE) methods are used to estimate the parameters in the model, and the plug-in method is introduced to obtain the estimation of conditional covariance matrix. Asymptotic properties are established for the proposed method, and simulation studies are given to demonstrate its performance. A financial application is presented to support the methodology.

scholarly journals Efficient computation of limit spectra of sample covariance matrices

2015 ◽  
Vol 04 (04) ◽  
pp. 1550019 ◽  
Author(s):  
Edgar Dobriban
Keyword(s):  
Covariance Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Discrete Distributions ◽  
High Dimensional ◽  
Efficient Computation ◽  
Usual Model ◽  
Sample Covariance ◽  
Distribution Of Eigenvalues

Models from random matrix theory (RMT) are increasingly used to gain insights into the behavior of statistical methods under high-dimensional asymptotics. However, the applicability of the framework is limited by numerical problems. Consider the usual model of multivariate statistics where the data is a sample from a multivariate distribution with a given covariance matrix. Under high-dimensional asymptotics, there is a deterministic map from the distribution of eigenvalues of the population covariance matrix (the population spectral distribution or PSD), to the of empirical spectral distribution (ESD). The current methods for computing this map are inefficient, and this limits the applicability of the theory. We propose a new method to compute numerically the ESD from an arbitrary input PSD. Our method, called SPECTRODE, finds the support and the density of the ESD to high precision; we prove this for finite discrete distributions. In computational experiments SPECTRODE outperforms existing methods by orders of magnitude in speed and accuracy. We apply it to compute expectations and contour integrals of the ESD, which are often central in applications. We also illustrate that SPECTRODE is directly useful in statistical problems, such as estimation and hypothesis testing for covariance matrices. Our proposal, implemented in open source software, may broaden the use of RMT in high-dimensional data analysis.

scholarly journals A note on the CLT of the LSS for sample covariance matrix from a spiked population model

2014 ◽  
Vol 130 ◽  
pp. 194-207 ◽  
Author(s):  
Qinwen Wang ◽  
Jack W. Silverstein ◽  
Jian-feng Yao
Keyword(s):  
Covariance Matrix ◽  
Population Model ◽  
Sample Covariance Matrix ◽  
Sample Covariance

scholarly journals Regularization for high-dimensional covariance matrix

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Xiangzhao Cui ◽  
Chun Li ◽  
Jine Zhao ◽  
Li Zeng ◽  
Defei Zhang ◽  
...  
Keyword(s):  
Covariance Matrix ◽  
Regularization Method ◽  
Moving Average ◽  
Covariance Structure ◽  
Real Data ◽  
High Dimensional ◽  
Underlying Structure ◽  
Frobenius Norm ◽  
Simulation Studies ◽  
Sample Covariance

AbstractIn many applications, high-dimensional problem may occur often for various reasons, for example, when the number of variables under consideration is much bigger than the sample size, i.e., p >> n. For highdimensional data, the underlying structures of certain covariance matrix estimates are usually blurred due to substantial random noises, which is an obstacle to draw statistical inferences. In this paper, we propose a method to identify the underlying covariance structure by regularizing a given/estimated covariance matrix so that the noises can be filtered. By choosing an optimal structure from a class of candidate structures for the covariance matrix, the regularization is made in terms of minimizing Frobenius-norm discrepancy. The candidate class considered here includes the structures of order-1 moving average, compound symmetry, order-1 autoregressive and order-1 autoregressive moving average. Very intensive simulation studies are conducted to assess the performance of the proposed regularization method for very high-dimensional covariance problem. The simulation studies also show that the sample covariance matrix, although performs very badly in covariance estimation for high-dimensional data, can be used to correctly identify the underlying structure of the covariance matrix. The approach is also applied to real data analysis, which shows that the proposed regularization method works well in practice.

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