Methodology and convergence rates for functional linear regression

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.

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)

References in corpus (1)

• Nonparametric methods for inference in the presence of instrumental variables

Cited by in corpus (42)

• Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains
• A reproducing kernel Hilbert space approach to functional linear regression
• Smoothing splines estimators for functional linear regression
• Functional data analysis for density functions by transformation to a Hilbert space
• Estimation in functional linear quantile regression
• Modelling and forecasting daily electricity load curves: a hybrid approach
• A Smooth and Locally Sparse Estimator for Functional Linear Regression via Functional SCAD Penalty
• A goodness-of-fit test for the functional linear model with scalar response
• Thresholding Projection Estimators in Functional Linear Models
• Nonparametric inference in generalized functional linear models
• Nonlinear manifold representations for functional data
• Functional linear regression via canonical analysis
• Factor models and variable selection in high-dimensional regression analysis
• Adaptive functional linear regression
• Varying-coefficient functional linear regression
• Intrinsic Riemannian Functional Data Analysis
• Functional linear regression with points of impact
• Wavelet-domain regression and predictive inference in psychiatric neuroimaging
• Estimation of functional derivatives
• Smooth Lasso Estimator for the Function-on-Function Linear Regression Model
• Asymptotic equivalence of functional linear regression and a white noise inverse problem
• Robust functional regression based on principal components
• A test of significance in functional quadratic regression
• Estimation in functional regression for general exponential families
• Feature Selection for Functional Data
• An autocovariance-based learning framework for high-dimensional functional time series
• Elastic Functional Principal Component Regression
• Functional time series prediction under partial observation of the future curve
• Conditional Independence Testing in Hilbert Spaces with Applications to Functional Data Analysis
• Honest Confidence Sets in Nonparametric IV Regression and Other Ill-Posed Models
• Adaptive estimation in circular functional linear models
• Optimal classification and nonparametric regression for functional data
• Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems
• On regularized polynomial functional regression
• Functional Lagged Regression with Sparse Noisy Observations
• An Alternative Approach to Functional Linear Partial Quantile Regression
• FAStEN: An Efficient Adaptive Method for Feature Selection and Estimation in High-Dimensional Functional Regressions
• Finite Sample Theory for High-Dimensional Functional/Scalar Time Series with Applications
• Consistently recovering the signal from noisy functional data
• Functional instrumental variable regression with an application to estimating the impact of immigration on native wages
• Parameter Regimes in Partial Functional Panel Regression
• The ET Interview: Professor Joel L. Horowitz