Selecting the Derivative of a Functional Covariate in Scalar-on-Function Regression

TL;DR

This paper develops statistical tests to select the most appropriate derivative of a functional covariate in scalar-on-function regression models, enhancing model specification and interpretation.

Contribution

It introduces formal testing procedures for choosing among different derivatives in scalar-on-function regression, including nested model tests and a J test for nonlinear models.

Findings

Tests effectively distinguish between derivatives in linear models.

Simulation studies confirm good finite-sample performance.

Application to chemometric data demonstrates practical utility.

Abstract

This paper presents tests to formally choose between regression models using different derivatives of a functional covariate in scalar-on-function regression. We demonstrate that for linear regression, models using different derivatives can be nested within a model that includes point-impact effects at the end-points of the observed functions. Contrasts can then be employed to test the specification of different derivatives. When nonlinear regression models are defined, we apply a $J$ test to determine the statistical significance of the nonlinear structure between a functional covariate and a scalar response. The finite-sample performance of these methods is verified in simulation, and their practical application is demonstrated using a chemometric data set.