Efficient robust nonparametric estimation in a semimartingale regression model

Victor Konev; Serguei Pergamenshchikov

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 4, page 1217-1244
  • ISSN: 0246-0203

Abstract

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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.

How to cite

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Konev, Victor, and Pergamenshchikov, Serguei. "Efficient robust nonparametric estimation in a semimartingale regression model." Annales de l'I.H.P. Probabilités et statistiques 48.4 (2012): 1217-1244. <http://eudml.org/doc/272010>.

@article{Konev2012,
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.},
author = {Konev, Victor, Pergamenshchikov, Serguei},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {non-asymptotic estimation; robust risk; model selection; sharp oracle inequality; asymptotic efficiency},
language = {eng},
number = {4},
pages = {1217-1244},
publisher = {Gauthier-Villars},
title = {Efficient robust nonparametric estimation in a semimartingale regression model},
url = {http://eudml.org/doc/272010},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Konev, Victor
AU - Pergamenshchikov, Serguei
TI - Efficient robust nonparametric estimation in a semimartingale regression model
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 4
SP - 1217
EP - 1244
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.
LA - eng
KW - non-asymptotic estimation; robust risk; model selection; sharp oracle inequality; asymptotic efficiency
UR - http://eudml.org/doc/272010
ER -

References

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