Asymptotically minimax nonparametric regression in $L_2$

Additive nonparametric regression with equidistant observation design is considered. The Pinsker-type minimax results are derived and the linear asymptotically minimax estimators are exhibited, based on approximation of the initial nonparametric model by a linear models of dimension which is increasing with the number of observations. The proof of optimality of these linear estimators within the class of all possible estimators is based on the rather elementary but very useful van Trees inequality. Keywords: Ellipsoid; asymptotic minimax risk; Bayes risk; asymptotically minimax estimator; second-order rate minimax estimator.