Wald Statistics in high-dimensional PCA^★

Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, CB3 0WB Cambridge, UK.

^** Corresponding author: m.loffler@statslab.cam.ac.uk

Received: 9 September 2018
Accepted: 10 January 2019

Abstract

In this study, we consider PCA for Gaussian observations X₁, …, X_n with covariance Σ = ∑_iλ_iP_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 projector ${\widehat{P}}_r$. 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 $\mathbf{\text{r}}(\Sigma )\gg \sqrt{n}$.