A note on estimation in Hilbertian linear models

TL;DR

This paper investigates estimation and prediction in Hilbert space-valued linear models, introducing a principal components estimator with minimal assumptions, applicable to time-dependent data including autoregressive Hilbertian models.

Contribution

It provides a consistent principal components based estimator for Hilbertian linear models under minimal assumptions, extending to time-dependent cases like autoregressive models.

Findings

Established consistency of the estimator under minimal conditions

Extended theory to time-dependent Hilbertian models

Applicable to autoregressive Hilbertian processes

Abstract

We study estimation and prediction in linear models where the response and the regressor variable both take values in some Hilbert space. Our main objective is to obtain consistency of a principal components based estimator for the regression operator under minimal assumptions. In particular, we avoid some inconvenient technical restrictions that have been used throughout the literature. We develop our theory in a time dependent setup which comprises as important special case the autoregressive Hilbertian model.