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

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  • Holger Dette

    (Ruhr-Universität Bochum)

  • Martin Kroll

    (Ruhr-Universität Bochum)

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. We derive sufficient conditions which imply asymptotic equivalence of the two models. We verify these conditions for the special cases of Sobolev ellipsoids and Hölder classes with smoothness index $$>1/2$$ > 1 / 2 under mild assumptions on the Gauss–Markov process. To give a counterexample, we 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 (see Brown and Low (Ann Stat 24:2384–2398, 1996)) can be extended to a setup with general Gauss–Markov noises.

Suggested Citation

  • Holger Dette & Martin Kroll, 2022. "Asymptotic equivalence for nonparametric regression with dependent errors: Gauss–Markov processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(6), pages 1163-1196, December.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:6:d:10.1007_s10463-022-00826-6
    DOI: 10.1007/s10463-022-00826-6
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    References listed on IDEAS

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    1. Iain M. Johnstone & Bernard W. Silverman, 1997. "Wavelet Threshold Estimators for Data with Correlated Noise," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 59(2), pages 319-351.
    2. Andrew V. Carter, 2009. "Asymptotically Sufficient Statistics in Nonparametric Regression Experiments with Correlated Noise," Journal of Probability and Statistics, Hindawi, vol. 2009, pages 1-19, January.
    3. Dette, Holger & Pepelyshev, Andrey & Zhigljavsky, Anatoly, 2016. "Optimal designs for regression models with autoregressive errors," Statistics & Probability Letters, Elsevier, vol. 116(C), pages 107-115.
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