  Semi-functional linear regression models postulate a linear relationship
between a scalar response and a functional covariate, and also include a
non-parametric component involving a univariate explanatory variable. It is of
practical importance to obtain estimators for these models that are robust
against high-leverage outliers, which are generally difficult to identify and
may cause serious damage to least squares and Huber-type $M$-estimators. For
that reason, robust estimators for semi-functional linear regression models are
constructed combining $B$-splines to approximate both the functional regression
parameter and the nonparametric component with robust regression estimators
based on a bounded loss function and a preliminary residual scale estimator.
Consistency and rates of convergence for the proposed estimators are derived
under mild regularity conditions. The reported numerical experiments show the
advantage of the proposed methodology over the classical least squares and
Huber-type $M$-estimators for finite samples. The analysis of real examples
illustrate that the robust estimators provide better predictions for
non-outlying points than the classical ones, and that when potential outliers
are removed from the training and test sets both methods behave very similarly.
