Efficient estimation of linear functionals of principal components
| Autor: | Vladimir Koltchinskii, Matthias Löffler, Richard Nickl |
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| Rok vydání: | 2020 |
| Předmět: |
Statistics and Probability
Trace (linear algebra) Rank (linear algebra) asymptotic normality Principal component analysis Asymptotic distribution Mathematics - Statistics Theory Statistics Theory (math.ST) 02 engineering and technology 01 natural sciences Combinatorics 010104 statistics & probability FOS: Mathematics 0202 electrical engineering electronic engineering information engineering 62H25 0101 mathematics Eigenvalues and eigenvectors Mathematics spectral projections Zero (complex analysis) Sigma 020206 networking & telecommunications semiparametric efficiency Covariance operator 62E17 Statistics Probability and Uncertainty Operator norm |
| Zdroj: | Ann. Statist. 48, no. 1 (2020), 464-490 |
| ISSN: | 0090-5364 |
| Popis: | 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. Comment: 48 pages, to appear in Annals of Statistics |
| Databáze: | OpenAIRE |
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