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Robust model selection for a semimartingale continuous time regression from discrete data

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  • Victor, Konev
  • Serguei, Pergamenchtchikov

Abstract

The paper considers the problem of estimating a periodic function in a continuous time regression model observed under a general semimartingale noise with an unknown distribution in the case when continuous observation cannot be provided and only discrete time measurements are available. Two specific types of noises are studied in detail: a non-Gaussian Ornstein–Uhlenbeck process and a time-varying linear combination of a Brownian motion and compound Poisson process. We develop new analytical tools to treat the adaptive estimation problems from discrete data. A lower bound for the frequency sampling, needed for the efficiency of the procedure constructed by discrete observations, has been found. Sharp non-asymptotic oracle inequalities for the robust quadratic risk have been derived. New convergence rates for the efficient procedures have been obtained. An example of the regression with a martingale noise exhibits that the minimax robust convergence rate may be both higher or lower as compared with the minimax rate for the “white noise” model. The results of Monte-Carlo simulations are given.

Suggested Citation

  • Victor, Konev & Serguei, Pergamenchtchikov, 2015. "Robust model selection for a semimartingale continuous time regression from discrete data," Stochastic Processes and their Applications, Elsevier, vol. 125(1), pages 294-326.
    • Handle: RePEc:eee:spapps:v:125:y:2015:i:1:p:294-326
    DOI: 10.1016/j.spa.2014.08.003
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    References listed on IDEAS

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    1. Victor Konev & Serguei Pergamenchtchikov, 2010. "General model selection estimation of a periodic regression with a Gaussian noise," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(6), pages 1083-1111, December.
    2. V. Konev & S. Pergamenshchikov, 2003. "Sequential Estimation of the Parameters in a Trigonometric Regression Model with the Gaussian Coloured Noise," Statistical Inference for Stochastic Processes, Springer, vol. 6(3), pages 215-235, October.
    3. Herold Dehling & Brice Franke & Thomas Kott, 2010. "Drift estimation for a periodic mean reversion process," Statistical Inference for Stochastic Processes, Springer, vol. 13(3), pages 175-192, October.
    4. 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.
    5. Galtchouk, L. & Pergamenshchikov, S., 2006. "Asymptotically efficient estimates for nonparametric regression models," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 852-860, April.
    6. Reinhard Höpfner & Yury Kutoyants, 2010. "Estimating discontinuous periodic signals in a time inhomogeneous diffusion," Statistical Inference for Stochastic Processes, Springer, vol. 13(3), pages 193-230, October.
    7. Comte, F. & Genon-Catalot, V. & Rozenholc, Y., 2009. "Nonparametric adaptive estimation for integrated diffusions," Stochastic Processes and their Applications, Elsevier, vol. 119(3), pages 811-834, March.
    8. 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.
    9. Ole E. Barndorff‐Nielsen & Neil Shephard, 2001. "Non‐Gaussian Ornstein–Uhlenbeck‐based models and some of their uses in financial economics," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 167-241.
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