Adaptive functional linear regression

TL;DR

This paper develops a fully data-driven method for estimating the slope function in functional linear regression, achieving near-optimal convergence rates without prior knowledge of key parameters.

Contribution

It introduces a new data-driven tuning parameter selection method combining model selection and Lepski's method for functional linear regression.

Findings

Achieves minimax-rates of convergence for various classes of functions.

Demonstrates effectiveness through simulation studies.

Applicable to prediction and derivative estimation in functional data.

Abstract

We consider the estimation of the slope function in functional linear regression, where scalar responses are modeled in dependence of random functions. Cardot and Johannes [J. Multivariate Anal. 101 (2010) 395-408] have shown that a thresholded projection estimator can attain up to a constant minimax-rates of convergence in a general framework which allows us to cover the prediction problem with respect to the mean squared prediction error as well as the estimation of the slope function and its derivatives. This estimation procedure, however, requires an optimal choice of a tuning parameter with regard to certain characteristics of the slope function and the covariance operator associated with the functional regressor. As this information is usually inaccessible in practice, we investigate a fully data-driven choice of the tuning parameter which combines model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski [Ann. Statist. 39 (2011) 1608-1632]. The tuning parameter is selected as minimizer of a stochastic penalized contrast function imitating Lepski's method among a random collection of admissible values. This choice of the tuning parameter depends only on the data and we show that within the general framework the resulting data-driven thresholded projection estimator can attain minimax-rates up to a constant over a variety of classes of slope functions and covariance operators. The results are illustrated considering different configurations which cover in particular the prediction problem as well as the estimation of the slope and its derivatives. A simulation study shows the reasonable performance of the fully data-driven estimation procedure.