Abstract: This study proposes a partial functional partially linear single-index model that consists of a functional linear component and a linear single-index component. This model generalizes many well-known existing models, and is suitable for more complicated data structures. We develop a new estimation procedure that combines a functional principal component analysis of the functional predictors, B-spline model for the parameters, and profile estimation of the unknown parameters and functions in the model. We establish the consistency and asymptotic normality of the parametric estimators. Furthermore, we derive the global convergence rate of the proposed estimator of the linear slope function, and establish that it is optimal in the minimax sense. We implement a two-stage procedure to estimate the nonparametric link function of the single-index component of the model; here, we find that the resulting estimator possesses the optimal global rate of convergence. Then, we obtain the convergence rate of the mean squared prediction error for a predictor. We study the empirical properties of the proposed procedures using Monte Carlo simulations. The proposed method is illustrated by analyzing a diffusion tensor imaging data set from the Alzheimer’s Disease Neuroimaging Initiative database.

Keywords: Asymptotic normality, consistency, functional data analysis, principal component analysis, single-index model.