Best estimation of functional linear models

TL;DR

This paper introduces a new estimation method for functional linear models that reconstructs response functions and derivatives from smoothed data, providing a more efficient estimator than traditional methods.

Contribution

It develops a strong functional Gauss-Markov theorem and proposes an estimator that leverages both response curves and derivatives for improved efficiency.

Findings

Simulation shows improved accuracy over traditional methods

Theoretical proof of estimator's efficiency

Estimation method applicable to various continuous process data

Abstract

Observations which are realizations from some continuous process are frequent in sciences, engineering, economics, and other fields. We consider linear models, with possible random effects, where the responses are random functions in a suitable Sobolev space. The processes cannot be observed directly. With smoothing procedures from the original data, both the response curves and their derivatives can be reconstructed, even separately. From both these samples of functions, just one sample of representatives is obtained to estimate the vector of functional parameters. A simulation study shows the benefits of this approach over the common method of using information either on curves or derivatives. The main theoretical result is a strong functional version of the Gauss-Markov theorem. This ensures that the proposed functional estimator is more efficient than the best linear unbiased estimator based only on curves or derivatives.