On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise

The experiments of nonparametric regression with equidistant design points and Gaussian white noise are considered. Brown and Low have proven asymptotic equivalence of these models under a quite general smoothness assumption on the parameter space of regression functions. In the present paper we focus on periodic Sobolev classes. We prove asymptotic equivalence of nonparametric regression and white noise with a construction different to Brown and Low. Whereas their original method cannot give a better rate than n−1/2 for the smoothness classes under consideration, even if the underlying function class is actually smoother than just Lipschitz, in the present work a rate of convergence n−β+1/2 for the delta-distance over a Sobolev class with any smoothness index β > 1/2 is derived. Furthermore, the results are constructive and therefore lead to a simple transfer of decision procedures.