Bibliographic Information: Gurdogan, H., & Shkolnik, A. (2024). The Quadratic Optimization Bias of Large Covariance Matrices. Annals of Statistics. (Manuscript submitted for publication).
Research Objective: This research paper investigates the interplay between optimization procedures and estimation errors in large covariance models, specifically focusing on the bias introduced when using plug-in estimators for quadratic optimization in high-dimensional settings.
Methodology: The authors analyze the asymptotic behavior of the discrepancy between true and realized quadratic optima, identifying a "quadratic optimization bias" function. They then develop a novel method for correcting the bias in sample eigenvectors obtained through PCA, leveraging a signal-to-noise ratio adjustment based on the Marchenko-Pastur distribution.
Key Findings: The study reveals that the accuracy of estimated eigenvectors, rather than eigenvalues, is crucial for minimizing the discrepancy in quadratic optimization. The proposed eigenvector correction method demonstrably reduces this discrepancy, leading to more accurate covariance estimates and improved performance in quadratic optimization tasks. Notably, the correction remains effective even when the number of spikes in the covariance matrix is greater than one, a scenario not addressed in prior work.
Main Conclusions: The paper highlights the limitations of standard PCA-based covariance estimation for quadratic optimization in high-dimensional, low sample size scenarios. It offers a practical solution through a novel eigenvector correction method, enhancing the accuracy of covariance estimates and downstream optimization results.
Significance: This research significantly contributes to the field of high-dimensional statistics and covariance estimation, particularly in its application to quadratic optimization problems prevalent in finance, signal processing, and other domains. The proposed eigenvector correction method addresses a critical gap in the literature, offering a more robust approach for handling large covariance matrices in practical settings.
Limitations and Future Research: The paper primarily focuses on linear growth of spiked eigenvalues with dimension, leaving room for exploration of alternative covariance models. Additionally, while the study establishes convergence, further research on convergence rates and the impact of misspecified spike numbers is warranted. Extending the proposed method to quadratic programming with inequality constraints presents another promising avenue for future work.
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by Hubeyb Gurdo... at arxiv.org 10-07-2024
https://arxiv.org/pdf/2410.03053.pdfDeeper Inquiries