A goodness-of-fit test for the functional linear model with scalar response

TL;DR

This paper introduces a new goodness-of-fit test for the functional linear model with scalar response, leveraging random projections and bootstrap calibration, with demonstrated effectiveness through simulations and real data applications.

Contribution

It extends a multivariate regression goodness-of-fit test to the functional data setting, providing a computationally simple and effective method.

Findings

Test performs well in finite samples across various basis types.

Bootstrap method effectively calibrates the test's distribution.

Applied successfully to real datasets for model validation.

Abstract

In this work, a goodness-of-fit test for the null hypothesis of a functional linear model with scalar response is proposed. The test is based on a generalization to the functional framework of a previous one, designed for the goodness-of-fit of regression models with multivariate covariates using random projections. The test statistic is easy to compute using geometrical and matrix arguments, and simple to calibrate in its distribution by a wild bootstrap on the residuals. The finite sample properties of the test are illustrated by a simulation study for several types of basis and under different alternatives. Finally, the test is applied to two datasets for checking the assumption of the functional linear model and a graphical tool is introduced. Supplementary materials are available online.