Edgeworth correction for the largest eigenvalue in a spiked PCA model

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.