Asymptotic equivalence for nonparametric regression with dependent errors: Gauss-Markov processes

DOI10.1007/S10463-022-00826-6arXiv2104.09485WikidataQ114227660 ScholiaQ114227660MaRDI QIDQ6365657

Holger Dette, Martin H. Kroll

Publication date: 19 April 2021

Abstract: For the class of Gauss-Markov processes we study the problem of asymptotic equivalence of the nonparametric regression model with errors given by the increments of the process and the continuous time model, where a whole path of a sum of a deterministic signal and the Gauss-Markov process can be observed. In particular we provide sufficient conditions such that asymptotic equivalence of the two models holds for functions from a given class, and we verify these for the special cases of Sobolev ellipsoids and H"older classes with smoothness index $>1/2$ under mild assumptions on the Gauss-Markov process at hand. To derive these results, we develop an explicit characterization of the reproducing kernel Hilbert space associated with the Gauss-Markov process, that hinges on a characterization of such processes by a property of the corresponding covariance kernel introduced by Doob. In order to demonstrate that the given assumptions on the Gauss-Markov process are in some sense sharp we also show that asymptotic equivalence fails to hold for the special case of Brownian bridge. Our findings demonstrate that the well-known asymptotic equivalence of the Gaussian white noise model and the nonparametric regression model with i.i.d. standard normal errors can be extended to a result treating general Gauss-Markov noises in a unified manner.

Mathematics Subject Classification ID

Statistics (62-XX)

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