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Model selection for the robust efficient signal processing observed with small Lévy noise

Author

Listed:
  • Slim Beltaief

    (Université de Rouen Normandie)

  • Oleg Chernoyarov

    (National Research University “Moscow Power Engineering Institute”)

  • Serguei Pergamenchtchikov

    (Université de Rouen Normandie)

Abstract

We develop a new model selection method for an adaptive robust efficient nonparametric signal estimation observed with impulse noise which is defined by a general non-Gaussian Lévy process. On the basis of the developed method, we construct estimation procedures which are analyzed in two settings: in non-asymptotic and in asymptotic ones. For the first time for such models, we show non-asymptotic sharp oracle inequalities for quadratic and robust risks, i.e., we show that the constructed procedures are optimal in the sense of sharp oracle inequalities. Next, by making use of the obtained oracle inequalities, we provide an asymptotic efficiency property for the developed estimation methods in an adaptive setting when the signal/noise ratio goes to infinity. We apply the developed model selection methods for the signal number detection problem in multi-path information transmission.

Suggested Citation

  • Slim Beltaief & Oleg Chernoyarov & Serguei Pergamenchtchikov, 2020. "Model selection for the robust efficient signal processing observed with small Lévy noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(5), pages 1205-1235, October.
    • Handle: RePEc:spr:aistmt:v:72:y:2020:i:5:d:10.1007_s10463-019-00726-2
    DOI: 10.1007/s10463-019-00726-2
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    References listed on IDEAS

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    1. L. Galtchouk & S. Pergamenshchikov, 2009. "Sharp non-asymptotic oracle inequalities for non-parametric heteroscedastic regression models," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(1), pages 1-18.
    2. D. Fourdrinier & S. Pergamenshchikov, 2007. "Improved Model Selection Method for a Regression Function with Dependent Noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 59(3), pages 435-464, September.
    3. Galtchouk, L. & Pergamenshchikov, S., 2006. "Asymptotically efficient estimates for nonparametric regression models," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 852-860, April.
    4. Evgeny Pchelintsev, 2013. "Improved estimation in a non-Gaussian parametric regression," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 15-28, April.
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    Cited by:

    1. Vlad Stefan Barbu & Slim Beltaief & Serguei Pergamenchtchikov, 2022. "Adaptive efficient estimation for generalized semi-Markov big data models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(5), pages 925-955, October.
    2. Evgeny Pchelintsev & Serguei Pergamenshchikov & Maria Povzun, 2022. "Efficient estimation methods for non-Gaussian regression models in continuous time," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(1), pages 113-142, February.

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