## The Annals of Statistics

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

#### 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.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 439-467.

Dates
Revised: February 2018
First available in Project Euclid: 30 November 2018

https://projecteuclid.org/euclid.aos/1543568594

Digital Object Identifier
doi:10.1214/18-AOS1693

Mathematical Reviews number (MathSciNet)
MR3909938

Zentralblatt MATH identifier
07036207

Subjects
Primary: 62G10: Hypothesis testing 62J05: Linear regression
Secondary: 62G09: Resampling methods

#### Citation

Cuesta-Albertos, Juan A.; García-Portugués, Eduardo; Febrero-Bande, Manuel; González-Manteiga, Wenceslao. Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes. Ann. Statist. 47 (2019), no. 1, 439--467. doi:10.1214/18-AOS1693. https://projecteuclid.org/euclid.aos/1543568594

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#### Supplemental materials

• Supplement to “Goodness-of-fit tests for the functional linear model based on randomly projected empirical processes”. Two extra Appendices contain the proofs of the technical lemmas and further results for the simulation study.