### Abstract

The paper considers the problem of robust estimating a periodic function in a continuous time regression model with the dependent disturbances given by a general square integrable semimartingale with an unknown distribution. An example of such a noise is a non-Gaussian Ornstein-Uhlenbeck process with jumps (see (J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167-241), (Ann. Appl. Probab. 18 (2008) 879-908)). An adaptive model selection procedure, based on the weighted least square estimates, is proposed. Under general moment conditions on the noise distribution, sharp non-asymptotic oracle inequalities for the robust risks have been derived and the robust efficiency of the model selection procedure has been shown. It is established that, in the case of the non-Gaussian Ornstein-Uhlenbeck noise, the sharp lower bound for the robust quadratic risk is determined by the limit value of the noise intensity at high frequencies. An example with a martinagale noise exhibits that the risk convergence rate becomes worse if the noise intensity is unbounded.

Original language | English |
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Pages (from-to) | 1217-1244 |

Number of pages | 28 |

Journal | Annales de l'institut Henri Poincare (B) Probability and Statistics |

Volume | 48 |

Issue number | 4 |

DOIs | |

Publication status | Published - Nov 2012 |

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### Keywords

- Asymptotic efficiency
- Model selection
- Non-asymptotic estimation
- Robust risk
- Sharp oracle inequality

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*48*(4), 1217-1244. https://doi.org/10.1214/12-AIHP488