Adaptive estimation of linear functionals in functional linear models

TL;DR

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

Contribution

It introduces a new adaptive tuning parameter selection method based on model selection and Lepski's method, improving practical applicability.

Findings

Achieves minimax optimal rates up to a logarithmic factor.

Applicable to point-wise and local average estimation of the slope.

Provides theoretical guarantees for the adaptive procedure.

Abstract

We consider the estimation of the value of a linear functional of the slope parameter in functional linear regression, where scalar responses are modeled in dependence of random functions. In Johannes and Schenk [2010] it has been shown that a plug-in estimator based on dimension reduction and additional thresholding can attain minimax optimal rates of convergence up to a constant. However, this estimation procedure 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 these are unknown in practice, we investigate a fully data-driven choice of the tuning parameter based on a combination of model selection and Lepski's method, which is inspired by the recent work of Goldenshluger and Lepski [2011]. The tuning parameter is selected as the minimizer of a stochastic penalized contrast function imitating Lepski's method among a random collection of admissible values. We show that this adaptive procedure attains the lower bound for the minimax risk up to a logarithmic factor over a wide range of classes of slope functions and covariance operators. In particular, our theory covers point-wise estimation as well as the estimation of local averages of the slope parameter.