Wald Statistics in high-dimensional PCA

Matthias Löffler

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arXiv · 1805.03839

Wald Statistics in high-dimensional PCA

Matthias Löffler

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Abstract

In this note we consider PCA for Gaussian observations $X_1,\dots, X_n$ with covariance $Σ=\sum_i λ_i P_i$ in the 'effective rank' setting with model complexity governed by $\mathbf{r}(Σ):=\text{tr}(Σ)/\| Σ\|$. 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 confidence ellipsoids and tests for spectral projectors $P_r$. Using higher order pertubation theory we are able to show that our Theorem remains valid even when $\mathbf{r}(Σ) \gg \sqrt{n}$.

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Matthias Löffler. 2018-05-10. Wald Statistics in high-dimensional PCA. https://arxiv.org/abs/1805.03839

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