# Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes

## TL;DR

This paper introduces computationally efficient goodness-of-fit tests for the functional linear model using randomly projected empirical processes, achieving root-n convergence and addressing high-dimensional challenges.

## Contribution

It proposes a novel testing approach based on random projections that simplifies computation and maintains statistical validity, with practical implementation and simulation validation.

## Key findings

- Tests exhibit root-n convergence rates.
- Method effectively handles high-dimensional functional data.
- Software implementation enables practical application.

## Abstract

We consider marked empirical processes indexed by a randomly projected functional covariate to construct goodness-of-fit tests for the functional linear model with scalar response. The test statistics are built from continuous functionals over the projected process, resulting in computationally efficient tests that exhibit root-n convergence rates and circumvent the curse of dimensionality. The weak convergence of the empirical process is obtained conditionally on a random direction, whilst the almost surely equivalence between the testing for significance expressed on the original and on the projected functional covariate is proved. The computation of the test in practice involves calibration by wild bootstrap resampling and the combination of several p-values, arising from different projections, by means of the false discovery rate method. The finite sample properties of the tests are illustrated in a simulation study for a variety of linear models, underlying processes, and alternatives. The software provided implements the tests and allows the replication of simulations and data applications.