Efficient estimation of linear functionals of principal components
Vladimir Koltchinskii, Matthias Löffler, Richard Nickl
The Annals of Statistics · 2020 · Volume 48 · Issue 1
Abstract
We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations $X_{1},\dots,X_{n}$ in a separable Hilbert space $\mathbb{H}$ with unknown covariance operator $\Sigma $. The complexity of the problem is characterized by its effective rank $\mathbf{r}(\Sigma):=\frac{\operatorname{tr}(\Sigma)}{\|\Sigma \|}$, where $\mathrm{tr}(\Sigma)$ denotes the trace of $\Sigma $ and $\|\Sigma\|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $\Sigma $. Under the assumption that $\mathbf{r}(\Sigma)=o(n)$, we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semiparametric optimality.
Keywords
62E17, 62H25, asymptotic normality, Principal Component Analysis, Semiparametric efficiency, spectral projections
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