Thresholding Projection Estimators in Functional Linear Models

Abstract

We consider the problem of estimating the regression function in functional linear regression models by proposing a new type of projection estimators which combine dimension reduction and thresholding. The introduction of a threshold rule allows to get consistency under broad assumptions as well as minimax rates of convergence under additional regularity hypotheses. We also consider the particular case of Sobolev spaces generated by the trigonometric basis which permits to get easily mean squared error of prediction as well as estimators of the derivatives of the regression function. We prove these estimators are minimax and rates of convergence are given for some particular cases.

Revised version for J. of Multivariate Analysis

References in corpus (6)

• Generalized functional linear models
• Methodology and convergence rates for functional linear regression
• Smoothing splines estimators for functional linear regression
• Nonlinear estimation for linear inverse problems with error in the operator
• On rate optimal local estimation in functional linear model
• Nonparametric estimation in functional linear models with second order stationary regressors

Cited by in corpus (15)

• Adaptive functional linear regression
• Functional linear regression with points of impact
• Asymptotic equivalence of functional linear regression and a white noise inverse problem
• Estimation in functional regression for general exponential families
• Adaptive estimation in circular functional linear models
• On rate optimal local estimation in functional linear model
• On rate optimal local estimation in nonparametric instrumental regression
• PCA-based estimation for functional linear regression with functional responses
• A note on the prediction error of principal component regression
• A simple method to construct confidence bands in functional linear regression
• A note on estimation in Hilbertian linear models
• Partially adaptive nonparametric instrumental regression
• Adaptive nonparametric estimation in the functional linear model with functional output
• Non-asymptotic Adaptive Prediction in Functional Linear Models
• Hybrid Regularisation of Functional Linear Models