# zbMATH — the first resource for mathematics

An additive nonparametric regression model is considered of the form $Y=\mu+r_1(x^1)+\dots+r_d(x^d)+\varepsilon,$ where $$r_i$$ are unknown functions to be estimated. A two-stage estimation procedure is proposed. At the first stage, projective estimates $$\tilde r_i$$ are derived. At the second stage multiplicative ($$\tilde r_i(x)\xi_i(x)$$) or additive ($$\tilde r_i(x)+\zeta_i(x)$$) improvements are made which minimize the (estimated) locally smoothed $$L_2$$ distances from the true $$r_i$$. It is shown that optimal rates of MSE convergence can be derived for these estimates, and for $$r_i$$ for some “good” functional spaces the estimates can achieve the “parametric” rate MSE$$\sim O(n^{-1})$$. A cross-validation algorithm is proposed for optimal bandwidth selection. Results of simulations are presented.