On rate optimal local estimation in functional linear model

TL;DR

This paper develops a minimax optimal estimator for linear functionals in functional linear regression, demonstrating its effectiveness through theoretical bounds and practical examples.

Contribution

It introduces a dimension reduction and thresholding based estimator that is proven to be minimax optimal for estimating linear functionals of the slope in functional linear models.

Findings

Estimator is consistent under mild assumptions.

Achieves the lower bound for mean squared error, proving minimax optimality.

Applicable to various functionals, including point-wise and average estimates.

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. The theory in this paper covers in particular point-wise estimation as well as the estimation of weighted averages of the slope parameter. We propose a plug-in estimator which is based on a dimension reduction technique and additional thresholding. It is shown that this estimator is consistent under mild assumptions. We derive a lower bound for the maximal mean squared error of any estimator over a certain ellipsoid of slope parameters and a certain class of covariance operators associated with the regressor. It is shown that the proposed estimator attains this lower bound up to a constant and hence it is minimax optimal. Our results are appropriate to discuss a wide range of possible regressors, slope parameters and functionals. They are illustrated by considering the point-wise estimation of the slope parameter or its derivatives and its average value over a given interval.