Nonparametric estimation in functional linear models with second order stationary regressors

TL;DR

This paper introduces a new orthogonal series estimator with thresholding for functional linear regression with stationary regressors, achieving minimax optimal convergence rates across various regularity spaces and covariance conditions.

Contribution

It proposes a novel thresholded orthogonal series estimator for the slope in functional linear models with stationary regressors, establishing its optimal convergence rates.

Findings

Estimator achieves minimax optimal rates.

Performance measured across diverse risk functions.

Illustrated with Sobolev spaces and smoothing covariance operators.

Abstract

We consider the problem of estimating the slope parameter in functional linear regression, where scalar responses Y1,...,Yn are modeled in dependence of second order stationary random functions X1,...,Xn. An orthogonal series estimator of the functional slope parameter with additional thresholding in the Fourier domain is proposed and its performance is measured with respect to a wide range of weighted risks covering as examples the mean squared prediction error and the mean integrated squared error for derivative estimation. In this paper the minimax optimal rate of convergence of the estimator is derived over a large class of different regularity spaces for the slope parameter and of different link conditions for the covariance operator. These general results are illustrated by the particular example of the well-known Sobolev space of periodic functions as regularity space for the slope parameter and the case of finitely or infinitely smoothing covariance operator.