Adaptive Dwo Estimator of a Regression Function

Abstract

We address a problem of non-parametric estimation of an unknown regression function f : [-1/2,1/2] → R at a fixed point x0 ∊ (-1/2,1/2) on the basis of observations (xi, yi), i =l,.., n such that yi = f(xi) + ei, where ei ~ N(0, σ2) is unobservable, Gaussian i.i.d. random noise and xi ∊ [-1/2, 1/2] are given design points. Recently, the Direct Weight Optimization (DWO) method has been proposed to solve a problem of such kind. The properties of the method have been studied for the case when the unknown function f is continuously differentiable with Lipschitz continuous derivative having a priori known Lipschitz constant L. The minimax optimality and adaptivity with respect to the design have been established for the resulting estimator. However, in order to implement the approach, both L and σ are to be known. The subject of the submission is the study of an adaptive version of the DWO estimator which uses a data-driven choice of the method parameter L.