Edgeworth correction for the largest eigenvalue in a spiked PCA model

2017/10/18 by Jeha Yang, Iain M. Johnstone, Yang, Jeha +1
Mathematics · #FOS: Mathematics #Random Matrices and Applications #Statistical Methods and Bayesian Inference #Statistics Theory (math.ST) #Stochastic processes and statistical mechanics #math.ST #stat.TH

paper · pdf · doi:10.48550/arxiv.1710.06899

arxiv created 2017/10/18 · openalex publication_date 2017/10/18 · arxiv updated 2017/10/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study improved approximations to the distribution of the largest eigenvalue ℓ of the sample covariance matrix of n zero-mean Gaussian observations in dimension p+1. We assume that one population principal component has variance ℓ > 1 and the remaining `noise' components have common variance 1. In the high dimensional limit p/n → γ> 0, we begin study of Edgeworth corrections to the limiting Gaussian distribution of ℓ in the supercritical case ℓ > 1 + √ γ. The skewness correction involves a quadratic polynomial as in classical settings, but the coefficients reflect the high dimensional structure. The methods involve Edgeworth expansions for sums of independent non-identically distributed variates obtained by conditioning on the sample noise eigenvalues, and limiting bulk properties and fluctuations of these noise eigenvalues.

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