Goodness-of-fit tests for functional linear models based on integrated projections

TL;DR

This paper introduces a new goodness-of-fit test for functional linear models with both functional and scalar responses, featuring an easy-to-compute statistic, bootstrap calibration, and a diagnostic tool, implemented in R.

Contribution

It extends existing scalar response tests to functional responses, providing a practical, visual, and computationally efficient method with an R package for reproducibility.

Findings

The test effectively detects deviations from the functional linear model.

The bootstrap calibration ensures accurate significance levels.

The R package goffda facilitates application and reproducibility.

Abstract

Functional linear models are one of the most fundamental tools to assess the relation between two random variables of a functional or scalar nature. This contribution proposes a goodness-of-fit test for the functional linear model with functional response that neatly adapts to functional/scalar responses/predictors. In particular, the new goodness-of-fit test extends a previous proposal for scalar response. The test statistic is based on a convenient regularized estimator, is easy to compute, and is calibrated through an efficient bootstrap resampling. A graphical diagnostic tool, useful to visualize the deviations from the model, is introduced and illustrated with a novel data application. The R package goffda implements the proposed methods and allows for the reproducibility of the data application.