Predictive Functional Linear Models with Diverging Number of Semiparametric Single-Index Interactions.

When predicting crop yield using both functional and multivariate predictors, the prediction performances benefit from the inclusion of the interactions between the two sets of predictors. We assume the interaction depends on a nonparametric, single-index structure of the multivariate predictor and reduce each functional predictor's dimension using functional principal component analysis (FPCA). Allowing the number of FPCA scores to diverge to infinity, we consider a sequence of semiparametric working models with a diverging number of predictors, which are FPCA scores with estimation errors. We show that the parametric component of the model is root-n consistent and asymptotically normal, the overall prediction error is dominated by the estimation of the nonparametric interaction function, and justify a CV-based procedure to select the tuning parameters.