Efficient Estimation of Linear Functionals of Principal Components

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 $\mathrm{\Sigma }\mathrm{.}$ The complexity of the problem is characterized by its effective rank ${\bf r}(\Sigma):= \frac{{\rm tr}(\Sigma)}{|\Sigma|},$where${\rm tr}(\Sigma)$denotesthetraceof$\Sigma$and$|\Sigma|$ denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of $\mathrm{\Sigma }\mathrm{.}$ Under the assumption that $\mathbf{r}(\mathrm{\Sigma })=o(n),$ we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semi-parametric optimality.