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December 2008 Statistical eigen-inference from large Wishart matrices
N. Raj Rao, James A. Mingo, Roland Speicher, Alan Edelman
Ann. Statist. 36(6): 2850-2885 (December 2008). DOI: 10.1214/07-AOS583

Abstract

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., “eigen-inference”).

Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically, we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure.

Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are “global” (in a sense that we describe) and exploit “global” information thereby overcoming the inherent biases that cripple classical inference procedures which are “local” and rely on “local” information.

Citation

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N. Raj Rao. James A. Mingo. Roland Speicher. Alan Edelman. "Statistical eigen-inference from large Wishart matrices." Ann. Statist. 36 (6) 2850 - 2885, December 2008. https://doi.org/10.1214/07-AOS583

Information

Published: December 2008
First available in Project Euclid: 5 January 2009

zbMATH: 1168.62056
MathSciNet: MR2485015
Digital Object Identifier: 10.1214/07-AOS583

Subjects:
Primary: 15A52 , 62510 , 62E20

Keywords: eigen-inference , Free probability , linear statistics , Random matrix theory , Sample covariance matrices , second order freeness , Wishart matrices

Rights: Copyright © 2008 Institute of Mathematical Statistics

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Vol.36 • No. 6 • December 2008
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