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

Eduardo García‐Portugués, Javier Álvarez‐Liébana, Gonzalo Álvarez‐Pérez and Wenceslao González‐Manteiga

Scandinavian Journal of Statistics, 2021, vol. 48, issue 2, 502-528

Abstract: 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.

Date: 2021
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