Locally sparse quantile estimation for a partially functional interaction model

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.