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

postprin

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

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

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

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

postprin

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

postprin

CLT for eigenvalue statistics of large-dimensional general Fisher matrices with applications

Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two covariance matrices, or testing the independence between sub-groups of a multivariate random vector. Most of the existing work on random Fisher matrices deals with a particular situation where the population covariance matrices are equal. In this paper, we consider general Fisher matrices with arbitrary population covariance matrices and develop their spectral properties when the dimensions are proportionally large compared to the sample size. The paper has two main contributions: first the limiting distribution of the eigenvalues of a general Fisher matrix is found and second, a central limit theorem is established for a wide class of functionals of these eigenvalues. Applications of the main results are also developed for testing hypotheses on high-dimensional covariance matrices.published_or_final_versio