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We consider marked empirical processes indexed by a randomly projectedfunctional covariate to construct goodness-of-fit tests for the functionallinear model with scalar response. The test statistics are built fromcontinuous functionals over the projected process, resulting in computationallyefficient tests that exhibit root-n convergence rates and circumvent the curseof dimensionality. The weak convergence of the empirical process is obtainedconditionally on a random direction, whilst the almost surely equivalencebetween the testing for significance expressed on the original and on theprojected functional covariate is proved. The computation of the test inpractice involves calibration by wild bootstrap resampling and the combinationof several p-values, arising from different projections, by means of the falsediscovery rate method. The finite sample properties of the tests areillustrated in a simulation study for a variety of linear models, underlyingprocesses, and alternatives. The software provided implements the tests andallows the replication of simulations and data applications.

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