Journal
ANNALS OF STATISTICS
Volume 35, Issue 1, Pages 70-91Publisher
INST MATHEMATICAL STATISTICS
DOI: 10.1214/009053606000000957
Keywords
deconvolution; dimension reduction; eigenfunction; eigenvalue; linear operator; minimax optimality; nonparametric; principal components analysis; smoothing; quadratic; regularisation
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Funding
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [0906795] Funding Source: National Science Foundation
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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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