  We consider the problem of variable selection in varying-coefficient
functional linear models, where multiple predictors are functions and a
response is a scalar and depends on an exogenous variable. The
varying-coefficient functional linear model is estimated by the penalized
maximum likelihood method with the sparsity-inducing penalty. Tuning parameters
that controls the degree of the penalization are determined by a model
selection criterion. The proposed method can reveal which combination of
functional predictors relates to the response, and furthermore how each
predictor relates to the response by investigating coefficient surfaces.
Simulation studies are provided to investigate the effectiveness of the
proposed method. We also apply it to the analysis of crop yield data to
investigate which combination of environmental factors relates to the amount of
a crop yield.
