Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process

Eva Löcherbach; Dasha Loukianova; Oleg Loukianov

ESAIM: Probability and Statistics (2012)

  • Volume: 15, page 197-216
  • ISSN: 1292-8100

Abstract

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Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist.47 (2011) 425–449].

How to cite

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Löcherbach, Eva, Loukianova, Dasha, and Loukianov, Oleg. "Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process." ESAIM: Probability and Statistics 15 (2012): 197-216. <http://eudml.org/doc/222472>.

@article{Löcherbach2012,
abstract = { Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist.47 (2011) 425–449].},
author = {Löcherbach, Eva, Loukianova, Dasha, Loukianov, Oleg},
journal = {ESAIM: Probability and Statistics},
keywords = {Diffusion process; adaptive estimation; regeneration method; mean square estimator; model selection; deviation inequalities; diffusion process},
language = {eng},
month = {1},
pages = {197-216},
publisher = {EDP Sciences},
title = {Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process},
url = {http://eudml.org/doc/222472},
volume = {15},
year = {2012},
}

TY - JOUR
AU - Löcherbach, Eva
AU - Loukianova, Dasha
AU - Loukianov, Oleg
TI - Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process
JO - ESAIM: Probability and Statistics
DA - 2012/1//
PB - EDP Sciences
VL - 15
SP - 197
EP - 216
AB - Let X be a one dimensional positive recurrent diffusion continuously observed on [0,t] . We consider a non parametric estimator of the drift function on a given interval. Our estimator, obtained using a penalized least square approach, belongs to a finite dimensional functional space, whose dimension is selected according to the data. The non-asymptotic risk-bound reaches the minimax optimal rate of convergence when t → ∞. The main point of our work is that we do not suppose the process to be in stationary regime neither to be exponentially β-mixing. This is possible thanks to the use of a new polynomial inequality in the ergodic theorem [E. Löcherbach, D. Loukianova and O. Loukianov, Ann. Inst. H. Poincaré Probab. Statist.47 (2011) 425–449].
LA - eng
KW - Diffusion process; adaptive estimation; regeneration method; mean square estimator; model selection; deviation inequalities; diffusion process
UR - http://eudml.org/doc/222472
ER -

References

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