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 $Sigma.$ The complexity of the problem is characterized by its effective rank ${bf r}(Sigma):= frac{{rm tr}(Sigma)}{|Sigma|},$ where ${rm tr}(Sigma)$ 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 ${bf r}(Sigma)=o(n),$ we establish the asymptotic normality and asymptotic properties of the risk of the resulting estimators and prove matching minimax lower bounds, showing their semi-parametric optimality.