Efficient estimation of linear functionals of principal components

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.