Wald Statistics in high-dimensional PCA

Abstract

In this note we consider PCA for Gaussian observations X 1 , . . . , Xn with covariance Σ = i λ i P i 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 projectorPr. This can be used to construct non-asymptotic confidence ellipsoids and tests for spectral projectors Pr. Using higher order pertubation theory we are able to show that our Theorem remains valid even when r(Σ) ≫ √ n.