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Title: Optimal prediction in generalized functional linear model with semiparametric single-index interactions Authors:  Yanghui Liu - East China Normal University (China) [presenting]
Yehua Li - University of California at Riverside (United States)
Naisyin Wang - Univ of Michigan (United States)
Raymond Carroll - Texas A and M University (United States)
Riquan Zhang - East China Normal University (China)
Abstract: A generalized functional linear model with semiparametric single-index interactions is considered. The response variable was assumed previously to depend on multiple covariates as well as on a finite number of features in the functional predictor. We incorporate all features of the functional predictor into our prediction model. We consider a two-stage estimation procedure, in which a regularized functional predictor based on functional principal component analysis is used. The asymptotic properties of our estimators and rate of prediction are derived. It is shown that, under mild regularity conditions, the parametric estimators are $\sqrt{n}$-consistent, and are asymptotically normal when the functional predictor is under-smoothed. Furthermore, the overall convergence rate of squared prediction error is dominated by the nonparametric link function in the single-index component part, while the prediction rate for the functional linear part attains the optimal rate for traditional functional linear model in the minimax sense. Our theory also suggests that it works well to use a K-fold cross-validation procedure to identify a range of suitable $m$. We note that within this range, the prediction errors are insensitive toward the choice of $m$ and satisfactory outcomes are achieved. The finite sample properties of our methods are further illustrated by a simulation study and a crop yield prediction application.