On linear regression models in infinite dimensional spaces with scalar response

TL;DR

This paper investigates the properties of least squares estimation in functional linear regression models within infinite-dimensional spaces, focusing on the effects of different projection choices and asymptotic behavior as subspace dimensions grow.

Contribution

It provides a detailed analysis of the inferential properties and asymptotic behavior of least squares estimators under various projection subspace choices in functional linear regression.

Findings

Different projection subspaces affect estimation properties

Asymptotic behavior varies with increasing subspace dimension

Highlights the importance of basis selection in functional regression

Abstract

In functional linear regression, the parameters estimation involves solving a non necessarily well-posed problem and it has points of contact with a range of methodologies, including statistical smoothing, deconvolution and projection on finite-dimensional subspaces. We discuss the standard approach based explicitly on functional principal components analysis, nevertheless the choice of the number of basis components remains something subjective and not always properly discussed and justified. In this work we discuss inferential properties of least square estimation in this context with different choices of projection subspaces, as well as we study asymptotic behaviour increasing the dimension of subspaces.