Functional average derivative regression

Single index model is studied in the framework of regression estimation involving functional data. The model combines low sensitivity to dimensional effects together with flexibility and easiness of interpretation. This paper proposes to construct
a functional adaptation of the average derivative method for estimating the functional index involved in the model. The convergence (with rate) of the estimator of the index is also derived based on asymptotic results on nonparametric functional derivatives estimation. In addition, it is shown that the nonlinear regression component of the model can be estimated at the standard univariate rate, being therefore insensitive to the infinite dimensionality of the data. These appealing theoretical features of the functional average derivative method are combined with easiness of implementation as shown through finite sample studies.