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

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2021-06-01
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Wiley
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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.
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Bootstrap, Cramér-von Mises statistic, Functional data, Regularization, Goodness-offFit
Bibliographic citation
García‐Portugués, E., Álvarez‐Liébana, J., Álvarez‐Pérez, G., & González‐Manteiga, W. (2021). A goodness‐of‐fit test for the functional linear model with functional response. Scandinavian Journal of Statistics, 48 (2), pp. 502-528.