The Annals of Statistics

Methodology and convergence rates for functional linear regression

Peter Hall and Joel L. Horowitz

Full-text: Open access

Abstract

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.

Article information

Source
Ann. Statist., Volume 35, Number 1 (2007), 70-91.

Dates
First available in Project Euclid: 6 June 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1181100181

Digital Object Identifier
doi:10.1214/009053606000000957

Mathematical Reviews number (MathSciNet)
MR2332269

Zentralblatt MATH identifier
1114.62048

Subjects
Primary: 62J05: Linear regression
Secondary: 62G20: Asymptotic properties

Keywords
Deconvolution dimension reduction eigenfunction eigenvalue linear operator minimax optimality nonparametric principal components analysis smoothing quadratic regularisation

Citation

Hall, Peter; Horowitz, Joel L. Methodology and convergence rates for functional linear regression. Ann. Statist. 35 (2007), no. 1, 70--91. doi:10.1214/009053606000000957. https://projecteuclid.org/euclid.aos/1181100181


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