Wald Statistics in high-dimensional PCA

In this study, we consider PCA for Gaussian observations X-1, horizontal ellipsis , X-n with covariance sigma = n-ary sumation (i)lambda P-i(i) in the 'effective rank' setting with model complexity governed by r(sigma) colon equals tr(sigma)/parallel to sigma parallel to. We prove a Berry-Essen type bound for a Wald Statistic of the spectral projector $\hat P_r$Pr. This can be used to construct non-asymptotic goodness of fit tests and confidence ellipsoids for spectral projectors P-r. Using higher order pertubation theory we are able to show that our Theorem remains valid even when r(Sigma) >> root n.