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February 2020 Efficient estimation of linear functionals of principal components
Vladimir Koltchinskii, Matthias Löffler, Richard Nickl
Ann. Statist. 48(1): 464-490 (February 2020). DOI: 10.1214/19-AOS1816

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.

Citation

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Vladimir Koltchinskii. Matthias Löffler. Richard Nickl. "Efficient estimation of linear functionals of principal components." Ann. Statist. 48 (1) 464 - 490, February 2020. https://doi.org/10.1214/19-AOS1816

Information

Received: 1 September 2017; Revised: 1 January 2019; Published: February 2020
First available in Project Euclid: 17 February 2020

zbMATH: 07196547
MathSciNet: MR4065170
Digital Object Identifier: 10.1214/19-AOS1816

Subjects:
Primary: 62H25
Secondary: 62E17

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.48 • No. 1 • February 2020
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