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

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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.

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Acknowledgements

The second and last authors are partly supported by the Ministry of Education and Science of the Russian Federation in the framework of the research Project No. 2.3208.2017/4.6. The last author is partly supported by the Russian Federal Professor program (Project No. 1.472.2016/1.4, Ministry of Education and Science).

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Authors and Affiliations

  1. Laboratoire de Mathématiques Raphael Salem, Université de Rouen Normandie, Avenue de l’Université, BP.12, Technopôle du Madrillet, Saint-Étienne-du-Rouvray, 76801, France

    Slim Beltaief

  2. International Laboratory SSP&QF of Tomsk State University and Department of Radio Engineering Devices and Antenna Systems, National Research University “Moscow Power Engineering Institute”, Krasnokazarmennaya 14, Moscow, 111250, Russia

    Oleg Chernoyarov

  3. International Laboratory SSP&QF of Tomsk State University and Laboratoire de Mathématiques Raphael Salem, Université de Rouen Normandie, Avenue de l’Université, BP.12, Technopôle du Madrillet, Saint-Étienne-du-Rouvray, 76801, France

    Serguei Pergamenchtchikov

Authors
  1. Slim Beltaief
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  2. Oleg Chernoyarov
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  3. Serguei Pergamenchtchikov
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Correspondence to Serguei Pergamenchtchikov.

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This work was supported by the RSF Grant 17-11-01049 (National Research Tomsk State University). The online version of this article contains supplementary material.

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Beltaief, S., Chernoyarov, O. & Pergamenchtchikov, S. Model selection for the robust efficient signal processing observed with small Lévy noise. Ann Inst Stat Math 72, 1205–1235 (2020). https://doi.org/10.1007/s10463-019-00726-2

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  • DOI: https://doi.org/10.1007/s10463-019-00726-2

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