A test of significance in functional quadratic regression

TL;DR

This paper introduces a new significance test for nonlinear components in functional quadratic regression models, utilizing principal component analysis, with proven asymptotic properties and demonstrated effectiveness through simulations and real data application.

Contribution

The paper develops a novel testing method for nonlinear terms in functional quadratic regression, extending the common linear model and establishing its asymptotic behavior.

Findings

Test shows good size and power in finite samples

Method successfully applied to real near-infrared spectral data

Provides a reliable tool for nonlinear significance testing in functional data

Abstract

We consider a quadratic functional regression model in which a scalar response depends on a functional predictor; the common functional linear model is a special case. We wish to test the significance of the nonlinear term in the model. We develop a testing method which is based on projecting the observations onto a suitably chosen finite dimensional space using functional principal component analysis. The asymptotic behavior of our testing procedure is established. A simulation study shows that the testing procedure has good size and power with finite sample sizes. We then apply our test to a data set provided by Tecator, which consists of near-infrared absorbance spectra and fat content of meat.