Annals of Statistics

Efficient estimation of linear functionals of principal components

Vladimir Koltchinskii, Matthias Löffler, and Richard Nickl

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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.

Article information

Ann. Statist., Volume 48, Number 1 (2020), 464-490.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 62E17: Approximations to distributions (nonasymptotic)

Principal component analysis spectral projections asymptotic normality semiparametric efficiency


Koltchinskii, Vladimir; Löffler, Matthias; Nickl, Richard. Efficient estimation of linear functionals of principal components. Ann. Statist. 48 (2020), no. 1, 464--490. doi:10.1214/19-AOS1816.

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Supplemental materials

  • Supplement to “Efficient estimation of linear functionals of principal components”. Supplementary information.