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

TL;DR

This paper establishes conditions under which nonparametric regression models with dependent Gauss-Markov errors are asymptotically equivalent to continuous path observation models, extending classical results to dependent noise.

Contribution

It provides a comprehensive characterization of asymptotic equivalence for Gauss-Markov processes, including explicit RKHS descriptions and sharp conditions for various classes of functions.

Findings

Asymptotic equivalence holds for Sobolev and Hölder classes with smoothness > 1/2.

Explicit RKHS characterization for Gauss-Markov processes is developed.

Asymptotic equivalence fails for Brownian bridge.

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.