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 |
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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
Efficient robust nonparametric estimation in a semimartingale regression model. / Konev, Victor; Pergamenshchikov, Serguei.
In: Annales de l'institut Henri Poincare (B) Probability and Statistics, Vol. 48, No. 4, 11.2012, p. 1217-1244.Research output: Contribution to journal › Article
}
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
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U2 - 10.1214/12-AIHP488
DO - 10.1214/12-AIHP488
M3 - Article
AN - SCOPUS:84879249535
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 -