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

Model must align true,
Goodness-of-fit shows the way
Functional beauty
Dimension reduction
Functional data
Functional linear models
R package
Elastic net regularization
Weather data

Eduardo García Portugués, Javier Álvarez Liébana, Gonzalo Álvarez Pérez and Wenceslao González Manteiga, «A goodness-of-fit test for the functional linear model with functional response», Scand. J. Statist. 48 (2), 502–528 (2021), doi: 10.1111/sjos.12486


Universidad Carlos III

Universidad Complutense de Madrid

Universidad de Oviedo

Universidad de Santiago de Compostela


June 2021

Otros detalles

Incluido como parte del Special Issue asociado al 4th Workshop on Goodness-of-Fit, Change-Point and Related Problems» celebrado en Trento en 2019



The functional linear model with functional response (FLMFR) is one of the most fundamental models to assess the relation between two functional random variables. In this article, we propose a novel goodness-of-fit test for the FLMFR against a general, unspecified, alternative. The test statistic is formulated in terms of a Cramér–von Mises norm over a doubly projected empirical process which, using geometrical arguments, yields an easy-to-compute weighted quadratic norm. A resampling procedure calibrates the test through a wild bootstrap on the residuals and the use of convenient computational procedures. As a sideways contribution, and since the statistic requires a reliable estimator of the FLMFR, we discuss and compare several regularized estimators, providing a new one specifically convenient for our test. The finite sample behavior of the test is illustrated via a simulation study. Also, the new proposal is compared with previous significance tests. Two novel real data sets illustrate the application of the new test.

Código R y datos

El código está documentado y libremente disponible en el repositorio GitHub. Además dicho software fue subido como paquete de R.

Hay dos archivos de datos incluidos en el repositorio:

Cita BibTeX

  author = {E. García-Portugués and J. Álvarez-Liébana and G. Álvarez-Pérez and W. González-Manteiga},
  title = {A goodness-of-fit test for the functional linear model with functional response},
  journal = {Scandinavian Journal of Statistics},
  volume = {48},
  number = {2},
  pages = {502-528},
  keywords = {bootstrap, Cramér–von Mises statistic, functional data, goodness-of-fit, regularization},
  doi = {10.1111/sjos.12486},
  url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/sjos.12486},
  year = {2021}