A k-nearest neighbor approach for functional regression

Let (X,Y) be a random pair taking values in , where is an infinite dimensional separable Hilbert space. We establish weak consistency of a nearest neighbor type estimator of the regression function of Y on X based on independent observations of the pair (X,Y). As a general strategy, we propose to reduce the infinite dimension of by considering only the first d coefficients of an expansion of X in an orthonormal system of , and then to perform k-nearest neighbor regression in . Both the dimension and the number of neighbors are automatically selected from the observations using a simple data-dependent splitting device.