Abstract: This study examines an estimating problem in single index models with functional predictors. An estimating approach is developed to estimate the slope function in the single-index and the nonparametric link function. Optimal convergence rates for the estimator of the slope function are established in a minimax sense, under mild conditions, using a functional principal component analysis and the estimating equation technique. For the estimator of the nonparametric link function, both the uniform and mean squared convergence rates are obtained. An error variance estimator is also defined and is proved to be asymptotically normal. The finite-sample performance of the proposed estimators is illustrated by simulations and a real-data application.

Key words and phrases: Functional data analysis, functional principal components analysis, kernel smoother, local linear smoothing, nonparametric models, semiparametric models.