EFFICIENT ESTIMATION OF LINEAR FUNCTIONALS OF PRINCIPAL COMPONENTS

• 成果类型: Article
• 署名作者: Koltchinskii, Vladimir; Loffler, Matthias; Nickl, Richard
• 署名单位: University System of Georgia; Georgia Institute of Technology; University of Cambridge
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/19-AOS1816
• 发表日期: 2020
• 页码: 464-490
• 关键词: confidence-intervals; spectral projectors; ASYMPTOTIC THEORY; bounds; eigenstructure; approximation; rates

摘要

We study principal component analysis (PCA) for mean zero i.i.d. Gaussian observations X-1, ..., X-n in a separable Hilbert space H with unknown covariance operator Sigma. The complexity of the problem is characterized by its effective rank r(Sigma) := tr(Sigma)/parallel to Sigma parallel to where tr(Sigma) denotes the trace of Sigma and parallel to Sigma parallel to 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 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.