Estimating functional single index models with compact support

The functional single index models are widely used to describe the nonlinear relationship between a scalar response and a functional predictor. The conventional functional single index model assumes that the coefficient function is nonzero in the entire time domain. In other words, the functional predictor always has a nonzero effect on the response all the time. We propose a new compact functional single index model, in which the coefficient function is only nonzero in a subregion. We also propose an efficient method that can simultaneously estimate the nonlinear link function, the coefficient function and also the nonzero region of the coefficient function. Hence, our method can identify the region in which the functional predictor is related to the response. Our method is illustrated by an application example in which the total number of daily bike rentals is predicted based on hourly temperature data. The finite sample performance of the proposed method is investigated by comparing it to the conventional functional single index model in a simulation study