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Efficient estimation methods for non-Gaussian regression models in continuous time

Author

Listed:
  • Evgeny Pchelintsev

    (Tomsk State University)

  • Serguei Pergamenshchikov

    (Université de Rouen)

  • Maria Povzun

    (Tomsk State University)

Abstract

In this paper, we develop an efficient nonparametric estimation theory for continuous time regression models with non-Gaussian Lévy noises in the case when the unknown functions belong to Sobolev ellipses. Using the Pinsker’s approach, we provide a sharp lower bound for the normalized asymptotic mean square accuracy. However, the main result obtained by Pinsker for the Gaussian white noise model is not correct without additional conditions for the ellipse coefficients. We find such constructive sufficient conditions under which we develop efficient estimation methods. We show that the obtained conditions hold for the ellipse coefficients of an exponential form. For exponential coefficients, the sharp lower bound is calculated in explicit form. Finally, we apply this result to signals number detection problems in multi-pass connection channels and we obtain an almost parametric convergence rate that is natural for this case, which significantly improves the rate with respect to power-form coefficients.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:1:d:10.1007_s10463-021-00790-7
    DOI: 10.1007/s10463-021-00790-7
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    References listed on IDEAS

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    1. E. A. Pchelintsev & S. M. Pergamenshchikov, 2018. "Oracle inequalities for the stochastic differential equations," Statistical Inference for Stochastic Processes, Springer, vol. 21(2), pages 469-483, July.
    2. P. Hodara & N. Krell & E. Löcherbach, 2018. "Non-parametric estimation of the spiking rate in systems of interacting neurons," Statistical Inference for Stochastic Processes, Springer, vol. 21(1), pages 81-111, April.
    3. Evgeny Pchelintsev, 2013. "Improved estimation in a non-Gaussian parametric regression," Statistical Inference for Stochastic Processes, Springer, vol. 16(1), pages 15-28, April.
    4. E. A. Pchelintsev & V. A. Pchelintsev & S. M. Pergamenshchikov, 2019. "Improved robust model selection methods for a Lévy nonparametric regression in continuous time," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 31(3), pages 612-628, July.
    5. 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.
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