Edgeworth correction for the largest eigenvalue in a spiked PCA model

TL;DR

This paper develops Edgeworth corrections to improve the approximation of the distribution of the largest eigenvalue in high-dimensional spiked PCA models, accounting for high-dimensional effects and skewness.

Contribution

It introduces Edgeworth correction formulas for the largest eigenvalue distribution in high-dimensional spiked PCA, incorporating high-dimensional structure and skewness.

Findings

Edgeworth corrections improve Gaussian approximation accuracy

Coefficients reflect high-dimensional structure

Method accounts for fluctuations of noise eigenvalues

Abstract

We study improved approximations to the distribution of the largest eigenvalue $\hat{\ell}$ of the sample covariance matrix of $n$ zero-mean Gaussian observations in dimension $p+1$. We assume that one population principal component has variance $\ell > 1$ and the remaining `noise' components have common variance $1$. In the high dimensional limit $p/n \to \gamma > 0$, we begin study of Edgeworth corrections to the limiting Gaussian distribution of $\hat{\ell}$ in the supercritical case $\ell > 1 + \sqrt \gamma$. 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 \textit{and} fluctuations of these noise eigenvalues.