Statistical Eigen-Inference from Large Wishart Matrices

Statistical Eigen-Inference from Large Wishart Matrices

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 i...

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Veröffentlicht in:The Annals of Statistics. - Institute of Mathematical Statistics. - 36(2008), 6, Seite 2850-2885
1. Verfasser: Rao, N. Raj (VerfasserIn)
Weitere Verfasser: Mingo, James A., Speicher, Roland, Edelman, Alan
Format: Online-Aufsatz
Sprache:English
Veröffentlicht: 2008
Zugriff auf das übergeordnete Werk:The Annals of Statistics
Schlagworte:Sample covariance matrices Random matrix theory Wishart matrices Second order freeness Free probability Eigen-inference Linear statistics Mathematics Social sciences Behavioral sciences
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520 |a 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. 
540 |a Copyright 2008 The Institute of Mathematical Statistics 
650 4 |a Sample covariance matrices 
650 4 |a Random matrix theory 
650 4 |a Wishart matrices 
650 4 |a Second order freeness 
650 4 |a Free probability 
650 4 |a Eigen-inference 
650 4 |a Linear statistics 
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650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Measures of variability  |x Multivariate statistical analysis  |x Covariance 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Vector analysis  |x Mathematical vectors  |x Eigenvectors 
650 4 |a Social sciences  |x Population studies  |x Population estimates 
650 4 |a Behavioral sciences  |x Psychology  |x Cognitive psychology  |x Cognitive processes  |x Thought processes  |x Reasoning  |x Inference 
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650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Statistical distributions  |x Distribution functions  |x Probability distributions  |x Mathematical moments 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Matrix theory  |x Eigenvalues 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Matrix theory  |x Matrices  |x Covariance matrices 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Measures of variability  |x Multivariate statistical analysis  |x Covariance 
650 4 |a Mathematics  |x Pure mathematics  |x Linear algebra  |x Vector analysis  |x Mathematical vectors  |x Eigenvectors 
650 4 |a Social sciences  |x Population studies  |x Population estimates 
650 4 |a Behavioral sciences  |x Psychology  |x Cognitive psychology  |x Cognitive processes  |x Thought processes  |x Reasoning  |x Inference 
650 4 |a Mathematics  |x Mathematical objects  |x Mathematical series  |x Power series 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Measures of variability  |x Sample size 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Statistical theories  |x Estimation theory  |x Estimation bias 
650 4 |a Mathematics  |x Applied mathematics  |x Statistics  |x Applied statistics  |x Descriptive statistics  |x Statistical distributions  |x Distribution functions  |x Probability distributions  |x Mathematical moments 
655 4 |a research-article 
700 1 |a Mingo, James A.  |e verfasserin  |4 aut 
700 1 |a Speicher, Roland  |e verfasserin  |4 aut 
700 1 |a Edelman, Alan  |e verfasserin  |4 aut 
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