A k-nearest neighbor approach for functional regression

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