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Efficient estimation of linear functionals of principal components. (English) Zbl 1440.62232
In the setting of principal component analysis for \(n\) IID, mean zero Gaussian observations in a separable Hilbert space, the authors consider the estimation problem for linear functionals of eigenvalues of the unknown covariance operator \(\Sigma\). The effective rank \(r(\Sigma)=\mbox{tr}(\Sigma)/\|\Sigma\|\) is used to quantify the complexity of the problem, where \(\mbox{tr}(\Sigma)\) is the trace of \(\Sigma\) and \(\|\Sigma\|\) is its operator norm. No assumptions on the structure of \(\Sigma\) are made, though eigenvalues to be estimated are assumed to be simple (i.e., have multiplicity 1). It is known that naive estimators can suffer from substantial bias when this effective rank is large with respect to \(n\). For the case where \(r(\Sigma)=o(n)\), the authors propose a bias reduction technique and show asymptotic normality of their estimator. Their upper bounds are complemented by lower bounds that demonstrate semiparametric optimality of their estimator in this case.

MSC:
62H25 Factor analysis and principal components; correspondence analysis
62E17 Approximations to statistical distributions (nonasymptotic)
62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
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References:
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