  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.
