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On the asymptotic equivalence and rate of convergence of nonparametric regression and Gaussian white noise. (English) Zbl 1057.62033
Summary: The experiments of nonparametric regression with equidistant design points and Gaussian white noise are considered. L. D. Brown and M. G. Low [Ann. Stat. 24, 2384–2398 (1996; Zbl 0867.62022)] have proven asymptotic equivalence of these models under a quite general smoothness assumption on the parameter space of regression functions. 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 \(n^{-\beta+1/2}\) for the delta-distance over a Sobolev class with any smoothness index \(\beta >1/2\) is derived. Furthermore, the results are constructive and therefore lead to a simple transfer of decision procedures.

62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference
62B15 Theory of statistical experiments
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