Efficient estimation of linear functionals of principal components

We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations X1,.,Xn in a separable Hilbert space H with unknown covariance operator .. The complexity of the problem is characterized by its effective rank r(.):=tr(.)..., where tr(.) denotes the trace of . and ... denotes its operator norm. We develop a method of bias reduction in the problem of estimation of linear functionals of eigenvectors of .. Under the assumption that r(.)=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.