Scalar-on-function local linear regression and beyond

It is common to want to regress a scalar response on a random function. This paper presents results that advocate local linear regression based on a projection as a nonparametric approach to this problem. Our asymptotic results demonstrate that functional local linear regression outperforms its functional local constant counterpart. Beyond the estimation of the regression operator itself, local linear regression is also a useful tool for predicting the functional derivative of the regression operator, a promising mathematical object on its own. The local linear estimator of the functional derivative is shown to be consistent. For both the estimator of the regression functional and the estimator of its derivative, theoretical properties are detailed. On simulated datasets we illustrate good finite sample properties of the proposed methods. On a real data example of a single-functional index model we indicate how the functional derivative of the regression operator provides an original, fast, and widely applicable estimation method.