EDGEWORTH CORRECTION FOR THE LARGEST EIGENVALUE IN A SPIKED PCA MODEL

Area of Science:

• Multivariate Statistics
• High-Dimensional Data Analysis
• Random Matrix Theory

Background:

• The distribution of the largest eigenvalue of sample covariance matrices is crucial in multivariate analysis.
• Classical approximations often fall short in high-dimensional settings where p/n approaches a constant.
• Understanding the impact of population principal component variances on eigenvalue distributions is key.

Purpose of the Study:

• To develop improved approximations for the largest eigenvalue distribution of sample covariance matrices.
• To investigate Edgeworth corrections in the supercritical regime where one component dominates.
• To analyze the influence of high-dimensional structure on statistical approximations.

Main Methods:

• Utilizing Edgeworth expansions for sums of independent, non-identically distributed random variables.
• Conditioning on sample noise eigenvalues to derive corrections.
• Analyzing the limiting bulk properties and fluctuations of noise eigenvalues in high dimensions.

Main Results:

• Derived Edgeworth corrections to the limiting Gaussian distribution of the largest eigenvalue.
• The skewness correction involves a quadratic polynomial whose coefficients are informed by the high-dimensional structure.
• Demonstrated the applicability of these methods in the supercritical case (ℓ > 1) and high-dimensional limit (p/n → γ > 0).

Conclusions:

• The proposed Edgeworth corrections offer more accurate approximations for the largest eigenvalue distribution.
• The findings highlight the importance of considering population structure in high-dimensional statistical inference.
• This work contributes to a deeper understanding of random matrix theory in practical data analysis scenarios.