A class of functional partially linear single-index models

The functional linear regression model is a useful extension of the classical linear model. However, it assumes a linear relationship between the response and functional covariates which may be invalid. For this reason, we generalize this model to a class of functional partially linear single-index models. In this paper, we propose a profile least squares approach combined with local constant smoothing for estimating the slope function and the link function in the new model. We demonstrate that our methods enable prediction of the link function and estimation of the slope function with polynomial convergence rates. The convergence rate of prediction of the whole model is also established. Monte Carlo simulation studies show an excellent finite-sample performance. A real data example about average yield of oats in Saskatchewan, Canada is used to illustrate our methodology.