Thresholding Projection Estimators in Functional Linear Models

TL;DR

This paper introduces a new projection estimator for functional linear regression that combines dimension reduction and thresholding, achieving consistency and minimax convergence rates under broad conditions.

Contribution

It proposes a novel thresholding projection estimator that improves estimation in functional linear models, with explicit results for Sobolev spaces and derivative estimation.

Findings

Estimator achieves minimax rates of convergence.

Consistent estimation under broad assumptions.

Effective in Sobolev space settings.

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.