Title: 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.