  A new sparse semiparametric model is proposed, which incorporates the
influence of two functional random variables in a scalar response in a flexible
and interpretable manner. One of the functional covariates is included through
a single-index structure, while the other is included linearly through the
high-dimensional vector formed by its discretised observations. For this model,
two new algorithms are presented for selecting relevant variables in the linear
part and estimating the model. Both procedures utilise the functional origin of
linear covariates. Finite sample experiments demonstrated the scope of
application of both algorithms: the first method is a fast algorithm that
provides a solution (without loss in predictive ability) for the significant
computational time required by standard variable selection methods for
estimating this model, and the second algorithm completes the set of relevant
linear covariates provided by the first, thus improving its predictive
efficiency. Some asymptotic results theoretically support both procedures. A
real data application demonstrated the applicability of the presented
methodology from a predictive perspective in terms of the interpretability of
outputs and low computational cost.
