Efficient Estimation of Linear Functionals of Principal Components

2017/08/25 by Koltchinskii, Vladimir, Löffler, Matthias, Nickl, Richard
#FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1708.07642

Abstract

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

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