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Jeha Yang1, Iain M Johnstone1

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This study refines approximations for the largest eigenvalue distribution in high-dimensional data. Improved Edgeworth corrections account for the structure of principal component variances in Gaussian samples.

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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.