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February 2007 Methodology and convergence rates for functional linear regression
Peter Hall, Joel L. Horowitz
Ann. Statist. 35(1): 70-91 (February 2007). DOI: 10.1214/009053606000000957


In functional linear regression, the slope “parameter” is a function. Therefore, in a nonparametric context, it is determined by an infinite number of unknowns. Its estimation involves solving an ill-posed problem and has points of contact with a range of methodologies, including statistical smoothing and deconvolution. The standard approach to estimating the slope function is based explicitly on functional principal components analysis and, consequently, on spectral decomposition in terms of eigenvalues and eigenfunctions. We discuss this approach in detail and show that in certain circumstances, optimal convergence rates are achieved by the PCA technique. An alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view.


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Peter Hall. Joel L. Horowitz. "Methodology and convergence rates for functional linear regression." Ann. Statist. 35 (1) 70 - 91, February 2007.


Published: February 2007
First available in Project Euclid: 6 June 2007

zbMATH: 1114.62048
MathSciNet: MR2332269
Digital Object Identifier: 10.1214/009053606000000957

Primary: 62J05
Secondary: 62G20

Keywords: Deconvolution , Dimension reduction , eigenfunction , eigenvalue , linear operator , Minimax optimality , nonparametric , principal components analysis , quadratic regularisation , smoothing

Rights: Copyright © 2007 Institute of Mathematical Statistics

Vol.35 • No. 1 • February 2007
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