Please use this identifier to cite or link to this item: http://scholarbank.nus.edu.sg/handle/10635/135450
Title: ON HIGH-DIMENSIONAL COVARIANCE MATRICES ESTIMATION
Authors: WEN JUN
Keywords: covariance matrix estimation,spectrum estimation,random matrix theory,Marčenko-Pastur equation,supersymmetry,Stieltjes transform
Issue Date: 6-Jan-2017
Source: WEN JUN (2017-01-06). ON HIGH-DIMENSIONAL COVARIANCE MATRICES ESTIMATION. ScholarBank@NUS Repository.
Abstract: It is widely accepted that covariance matrices play vital roles in various statistical problems. However, in most real-life applications, the population covariance matrix is unknown. Therefore, a good estimate of it is much in demand. It is known that when the magnitude of sample size is comparable to the dimension of the covariance matrix, conventional estimators no longer perform well. In this thesis, assuming large p, large n, we propose two estimators for the population covariance matrices. One deals with estimating a pair of covariance matrices $(\Sigma_{1}, \Sigma_{2})$ as well as the spectrum of $\Sigma_{2}\Sigma_{1}^{-1}$. The other focuses on estimating the spectrum of one covariance matrix $\Sigma$ only. For the former problem, we make use of the so-called Marčenko-Pastur equation technique, which associates the limiting spectral distribution of the ratio of sample covariance matrices $S_{2}S_{1}^{-1}$ to the one of $\Sigma_{2}\Sigma_{1}^{-1}$. For the latter problem, we utilize a method originated from physics, called Supersymmetry. By using anticommuting (Grassmann) variables, this method facilitates the computation of our optimization problem. Extensive Monte Carlo simulations indicate that our estimators have good finite sample performance.
URI: http://scholarbank.nus.edu.sg/handle/10635/135450
Appears in Collections:Ph.D Theses (Open)

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