Wald Statistics in high-dimensional PCA

Abstract

In this study, we consider PCA for Gaussian observations X1, …, Xn with covariance Σ = ∑iλiPi in the ’effective rank’ setting with model complexity governed by r(Σ) ≔ tr(Σ)∕∥Σ∥. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$. This can be used to construct non-asymptotic goodness of fit tests and confidence ellipsoids for spectral projectors Pr. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(\Sigma) \gg \sqrt{n}$.