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

Eva Löcherbach; Dasha Loukianova; Oleg Loukianov

ESAIM: Probability and Statistics (2011)

  • 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 (2011): 197-216. <http://eudml.org/doc/277136>.

@article{Löcherbach2011,
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},
language = {eng},
pages = {197-216},
publisher = {EDP-Sciences},
title = {Penalized nonparametric drift estimation for a continuously observed one-dimensional diffusion process},
url = {http://eudml.org/doc/277136},
volume = {15},
year = {2011},
}

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
PY - 2011
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
UR - http://eudml.org/doc/277136
ER -

References

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  1. [1] G. Banon, Nonparametric identification for diffusion processes. SIAM J. Control. Optim.16 (1978) 380–395. Zbl0404.93045MR492159
  2. [2] Y. Baraud, F. Comte and G. Viennet, Adaptive estimation in autoregression or β-mixing regression via model selection. Ann. Stat.29 (2001) 839–875. Zbl1012.62034MR1865343
  3. [3] Y. Baraud, F. Comte and G. Viennet, Model selection for (auto-)regression with dependent data. ESAIM: P&S 5 (2001) 33–49. Zbl0990.62035MR1845321
  4. [4] A. Barron, L. Birgé and P. Massart, Risks bounds for model selection via penalization. Prob. Th. Rel. Fields113 (1999) 301–413. Zbl0946.62036MR1679028
  5. [5] F. Comte, V. Genon-Catalot and Y. Rozenholc, Penalized nonparametric mean square estimation of the coefficients of diffusion processes. Bernoulli13 (2007) 514–543. Zbl1127.62067MR2331262
  6. [6] A. Dalalyan, Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Stat.33 (2005) 507–2528. Zbl1084.62079MR2253093
  7. [7] A.S. Dalalyan and Yu.A. Kutoyants, Asymptotically efficient trend coefficient estimation for ergodic diffusion. Math. Meth. Stat.11 (2002) 402–427. MR1979742
  8. [8] S. Delattre, M. Hoffmann and M. Kessler, Dynamics adaptive estimation of a scalar diffusion. Prpublication PMA-762, Univ. Paris 6 (2002). Available at www.proba.jussieu.fr/mathdoc/preprints/. Mathematical Reviews (MathSciNet): MR1895888 Project Euclid: euclid.bj/1078866865. MR1895888
  9. [9] L. Galtchouk and S. Pergamenschikov, Sequential nonparametric adaptive estimation of the drift coefficient in the diffusion processes. Math. Meth. Stat.10 (2001) 316–330. Zbl1005.62070MR1867163
  10. [10] M. Hoffmann, Adaptive estimation in diffusion processes. Stochastic Processes Appl.79 (1999) 135–163. Zbl1043.62528MR1670522
  11. [11] Yury A. Kutoyants, Statistical inference for ergodic diffusion processes. Springer Series in Statistics. London: Springer (2004). Zbl1038.62073MR2144185
  12. [12] O. Lepskii, One problem of adaptive estimation in Gaussian white noise. Theory Probab. Appl.35 (1999) 459–470. Zbl0745.62083MR1091202
  13. [13] G.G. Lorentz, M. von Golitschek and Y. Makovoz, Constructive approximation: advanced problems. Grundlehren der Mathematischen Wissenschaften 304. Berlin: Springer (1996). Zbl0910.41001MR1393437
  14. [14] D. Loukianova and O. Loukianov, Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions. Ann. Inst. Henri Poincaré44 (2008) 771–786. Zbl1182.62166MR2446297
  15. [15] E. Löcherbach and D. Loukianova, On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions. Stoch. Proc. Appl.118 (2008) 301–1321. Zbl1202.60122MR2427041
  16. [16] E. Löcherbach, D. Loukianova and O. Loukianov, Polynomial bounds in the Ergodic theorem for one-dimensional diffusions and integrability of hitting times. Ann. Inst. H. Poincaré Probab. Statist.47 (2011) 425–449. Zbl1220.60045MR2814417
  17. [17] T.D. Pham, Nonparametric estimation of the drift coefficient in the diffusion equation. Math. Operationsforsch. Statist., Ser. Statistics 1 (1981) 61–73. Zbl0485.62089MR607955
  18. [18] B.L.S. Prakasa Rao, Statistical Inference for Diffusion Type Processes. London: Edward Arnold. MR1717690 (1999) Zbl0952.62077MR1717690
  19. [19] D. Revuz and M. Yor, Continuous martingales and Brownian motion. 3rd ed. Grundlehren der Mathematischen Wissenschaften 293. Berlin: Springer (2005). Zbl1087.60040
  20. [20] V.G. Spokoiny, Adaptive drift estimation for nonparametric diffusion model. Ann. Stat.28 (2000) 815–836. Zbl1105.62330MR1792788
  21. [21] A.Yu. Veretennikov, On polynomial mixing bounds for stochastic differential equations. Stoch. Proc. Appl.70 (1997) 115–127. Zbl0911.60042MR1472961

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