Methodology and convergence rates for functional linear regression

2007/02/01 by Peter Hall, Joel L. Horowitz · 2 citations
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Statistical Methods and Inference #Stochastic Gradient Optimization Techniques #math.ST #msc:62G20 #msc:62J05 #stat.TH

paper · pdf · doi:10.1214/009053606000000957

published as Annals of Statistics 2007, Vol. 35, No. 1, 70-91 · Published at http://dx.doi.org/10.1214/009053606000000957 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

openalex publication_date 2007/02/01 · arxiv created 2007/08/03 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28

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.

Citations

• Instrumental Variable Estimation of Nonparametric Models
• Nonparametric methods for inference in the presence of instrumental variables
• On the Optimal Rates of Convergence for Nonparametric Deconvolution Problems
• Nonparametric Instrumental Regression

Cited by

• Smoothing splines estimators for functional linear regression
• Consistently recovering the signal from noisy functional data