[Paper Review] A note on the CLT of the LSS for sample covariance matrix from a spiked population model

Qinwen Wang, Jack W. Silverstein+1 more|arXiv (Cornell University)|Apr 23, 2013

Random Matrices and Applications18 references4 citations

TL;DR

This paper derives an asymptotic expansion for the centering parameter in the central limit theorem (CLT) for linear spectral statistics (LSS) of large-dimensional sample covariance matrices under a spiked population model. The key contribution is a refined centering term that accounts for multiple spike eigenvalues, enabling improved asymptotic power analysis for likelihood ratio tests on spike detection in high-dimensional data.

ABSTRACT

In this note, we establish an asymptotic expansion for the centering parameter appearing in the central limit theorems for linear spectral statistic of large-dimensional sample covariance matrices when the population has a spiked covariance structure. As an application, we provide an asymptotic power function for the corrected likelihood ratio statistic for testing the presence of spike eigenvalues in the population covariance matrix. This result generalizes an existing formula from the literature where only one simple spike exists.

Motivation & Objective

• To address the lack of explicit centering parameter formulas in CLT for linear spectral statistics (LSS) of sample covariance matrices in high-dimensional spiked population models.
• To extend existing CLT results—previously limited to single spikes—to the case of multiple, fixed-rank spikes in the population covariance matrix.
• To provide a precise asymptotic expansion of the centering parameter that captures the influence of multiple spike eigenvalues on the limiting distribution of LSS.
• To apply the refined centering term to derive an asymptotic power function for the corrected likelihood ratio test (LRT) for detecting spike eigenvalues.
• To generalize prior results that were restricted to a single spike, thereby broadening applicability to real-world factor models and high-dimensional statistical inference.

Proposed method

• Uses complex analysis and contour integration to derive the centering parameter in the CLT for LSS of sample covariance matrices under the spiked population model.
• Applies the Stieltjes transform and resolvent techniques to analyze the spectral distribution of the sample covariance matrix $ S_n $, particularly focusing on eigenvalue fluctuations.
• Employs a contour integral representation of the centering term involving the resolvent $ R(z) = (S_n - zI)^{-1} $, with integration over a contour $ abla $ enclosing the support of the limiting spectral distribution.
• Derives an asymptotic expansion of the centering parameter by computing residues at poles arising from the spike eigenvalues $ a_i $ and the Marçenko-Pastur law.
• Introduces a decomposition of the centering term into contributions from poles at $ m = -1 $, $ m = -1/a_i $, and $ m = 1/(y_n - 1) $, using residue calculus.
• Combines results from complex integration with known asymptotic expressions for the Stieltjes transform of the Marçenko-Pastur law to derive the final expansion.

Experimental results

Research questions

• RQ1 How does the centering parameter in the CLT for linear spectral statistics of sample covariance matrices depend on multiple spike eigenvalues in a high-dimensional spiked population model?
• RQ2 Can an explicit asymptotic expansion be derived for the centering term that accounts for the presence of multiple spikes, beyond the single-spike case?
• RQ3 What is the impact of multiple spikes on the limiting distribution of linear spectral statistics in large-dimensional settings?
• RQ4 How can the refined centering parameter be used to improve the asymptotic power analysis of the corrected likelihood ratio test for spike detection?
• RQ5 To what extent does the proposed expansion generalize existing results that assume only one spike eigenvalue?

Key findings

• The paper derives an asymptotic expansion for the centering parameter in the CLT for LSS of sample covariance matrices under a spiked population model with multiple spikes.
• The expansion explicitly accounts for the contributions of $ k $ spike eigenvalues $ a_i $ with multiplicities $ n_i $, extending prior results limited to a single spike.
• The centering parameter is expressed as a sum of residue terms involving $ a_i $, $ y_n = p/n $, and $ n_i $, with asymptotic error $ O(1/n^2) $.
• The derived centering term enables the construction of an asymptotic power function for the corrected likelihood ratio test (LRT) for spike detection.
• The power function generalizes previous formulas that were valid only for a single spike, now allowing inference in models with multiple dominant factors.
• The result confirms that the presence of multiple spikes significantly alters the centering of the limiting distribution of LSS, necessitating a refined theoretical treatment beyond the null case.