### 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 |
---|---|

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 |

### Fingerprint

### 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

**Efficient robust nonparametric estimation in a semimartingale regression model.** / Konev, Victor; Pergamenshchikov, Serguei.

Research output: Contribution to journal › Article

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 48, no. 4, pp. 1217-1244. https://doi.org/10.1214/12-AIHP488

}

TY - JOUR

T1 - Efficient robust nonparametric estimation in a semimartingale regression model

AU - Konev, Victor

AU - Pergamenshchikov, Serguei

PY - 2012/11

Y1 - 2012/11

N2 - 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.

AB - 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.

KW - Asymptotic efficiency

KW - Model selection

KW - Non-asymptotic estimation

KW - Robust risk

KW - Sharp oracle inequality

UR - http://www.scopus.com/inward/record.url?scp=84879249535&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84879249535&partnerID=8YFLogxK

U2 - 10.1214/12-AIHP488

DO - 10.1214/12-AIHP488

M3 - Article

VL - 48

SP - 1217

EP - 1244

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 4

ER -