On generalized expectation-based estimation of a population spectral distribution from high-dimensional data

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Extreme eigenvalues of large-dimensional spiked Fisher matrices with application

Consider two pp-variate populations, not necessarily Gaussian, with covariance matrices Σ1Σ1 and Σ2Σ2, respectively. Let S1S1 and S2S2 be the corresponding sample covariance matrices with degrees of freedom mm and nn. When the difference ΔΔ between Σ1Σ1 and Σ2Σ2 is of small rank compared to p,mp,m and nn, the Fisher matrix S:=S−12S1S:=S2−1S1 is called a spiked Fisher matrix. When p,mp,m and nn grow to infinity proportionally, we establish a phase transition for the extreme eigenvalues of the Fisher matrix: a displacement formula showing that when the eigenvalues of ΔΔ (spikes) are above (or under) a critical value, the associated extreme eigenvalues of SS will converge to some point outside the support of the global limit (LSD) of other eigenvalues (become outliers); otherwise, they will converge to the edge points of the LSD. Furthermore, we derive central limit theorems for those outlier eigenvalues of SS. The limiting distributions are found to be Gaussian if and only if the corresponding population spike eigenvalues in ΔΔ are simple. Two applications are introduced. The first application uses the largest eigenvalue of the Fisher matrix to test the equality between two high-dimensional covariance matrices, and explicit power function is found under the spiked alternative. The second application is in the field of signal detection, where an estimator for the number of signals is proposed while the covariance structure of the noise is arbitrary.published_or_final_versio

Moment approach for singular values distribution of a large auto-covariance matrix

Let (εt)t>0(εt)t>0 be a sequence of independent real random vectors of pp-dimension and let XT=∑s+Tt=s+1εtε∗t−s/TXT=∑t=s+1s+Tεtεt−s∗/T be the lag-ss (ss is a fixed positive integer) auto-covariance matrix of εtεt. Since XTXT is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of XTX∗TXTXT∗. Using the method of moments, we are able to investigate the limiting behaviors of the eigenvalues of XTX∗TXTXT∗ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit FF, which is a result previously developed in (J. Multivariate Anal. 137 (2015) 119–140) using the Stieltjes transform method. Second, we establish the convergence of its largest eigenvalue to the right edge of FF.published_or_final_versio

A local moments estimation of the spectrum of a large dimensional covariance matrix

This paper considers the problem of estimating the population spectral distribution from a sample covariance matrix when its dimension is large. We generalize the contour-integral based method in Mestre (2008) and present a local moment estimation procedure. Compared with the original, the new procedure can be applied successfully to models where the asymptotic clusters of sample eigenvalues generated by different population eigenvalues are not all separate. The proposed estimates are proved to be consistent. Numerical results illustrate the implementation of the estimation procedure and demonstrate its efficiency in various cases.postprin

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

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CLT for linear spectral statistics of a rescaled sample precision matrix

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Electrically controllable surface magnetism on the surface of topological insulator

We study theoretically the RKKY interaction between magnetic impurities on the surface of three-dimensional topological insulators, mediated by the helical Dirac electrons. Exact analytical expression shows that the RKKY interaction consists of the Heisenberg-like, Ising-like and DM-like terms. It provides us a new way to control surface magnetism electrically. The gap opened by doped magnetic ions can lead to a short-range Bloembergen-Rowland interaction. The competition among the Heisenberg, Ising and DM terms leads to rich spin configurations and anomalous Hall effect on different lattices.Comment: 5 pages, 3 figures, 1 tabl

A new multivariate CUSUM chart using principal components with a revision of Crosier's chart

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