  Functional data analysis has been extensively conducted. In this study, we
consider a partially functional model, under which some covariates are scalars
and have linear effects, while some other variables are functional and have
unspecified nonlinear effects. Significantly advancing from the existing
literature, we consider a model with interactions between the functional and
scalar covariates. To accommodate long-tailed error distributions which are not
uncommon in data analysis, we adopt the quantile technique for estimation. To
achieve more interpretable estimation, and to accommodate many practical
settings, we assume that the functional covariate effects are locally sparse
(that is, there exist subregions on which the effects are exactly zero), which
naturally leads to a variable/model selection problem. We propose respecting
the "main effect, interaction" hierarchy, which postulates that if a subregion
has a nonzero effect in an interaction term, then its effect has to be nonzero
in the corresponding main functional effect. For estimation, identification of
local sparsity, and respect of the hierarchy, we propose a penalization
approach. An effective computational algorithm is developed, and the
consistency properties are rigorously established under mild regularity
conditions. Simulation shows the practical effectiveness of the proposed
approach. The analysis of the Tecator data further demonstrates its practical
applicability. Overall, this study can deliver a novel and practically useful
model and a statistically and numerically satisfactory estimation approach.
