Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions

D. Loukianova; O. Loukianov

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 4, page 771-786
  • ISSN: 0246-0203

Abstract

top
Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya–Watson estimator in the case of a locally Hölder-continuous drift.

How to cite

top

Loukianova, D., and Loukianov, O.. "Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions." Annales de l'I.H.P. Probabilités et statistiques 44.4 (2008): 771-786. <http://eudml.org/doc/77991>.

@article{Loukianova2008,
abstract = {Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya–Watson estimator in the case of a locally Hölder-continuous drift.},
author = {Loukianova, D., Loukianov, O.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {Harris recurrence; diffusion processes; limit theorems; additive functionals; non-parametric estimation; Nadaraya–Watson estimator; rate of convergence; Nadaraya-Watson estimator},
language = {eng},
number = {4},
pages = {771-786},
publisher = {Gauthier-Villars},
title = {Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions},
url = {http://eudml.org/doc/77991},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Loukianova, D.
AU - Loukianov, O.
TI - Uniform deterministic equivalent of additive functionals and non-parametric drift estimation for one-dimensional recurrent diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 4
SP - 771
EP - 786
AB - Usually the problem of drift estimation for a diffusion process is considered under the hypothesis of ergodicity. It is less often considered under the hypothesis of null-recurrence, simply because there are fewer limit theorems and existing ones do not apply to the whole null-recurrent class. The aim of this paper is to provide some limit theorems for additive functionals and martingales of a general (ergodic or null) recurrent diffusion which would allow us to have a somewhat unified approach to the problem of non-parametric kernel drift estimation in the one-dimensional recurrent case. As a particular example we obtain the rate of convergence of the Nadaraya–Watson estimator in the case of a locally Hölder-continuous drift.
LA - eng
KW - Harris recurrence; diffusion processes; limit theorems; additive functionals; non-parametric estimation; Nadaraya–Watson estimator; rate of convergence; Nadaraya-Watson estimator
UR - http://eudml.org/doc/77991
ER -

References

top
  1. [1] A. Borodin and P. Salminen. Handbook of Brownian Motion – Facts and Formulae. Probability and its Applications. Birkhäuser, Basel, 1996. Zbl0859.60001MR1477407
  2. [2] M. Brancovan. Fonctionnelles additives spéciales des processus récurrents au sens de Harris. Z. Wahrsch. Verw. Gebiete 47 (1979) 163–194. Zbl0381.60065MR523168
  3. [3] X. Chen. How often does a Harris recurrent Markov chain recur? Ann. Probab. 27 (1999) 1324–1346. Zbl0981.60023MR1733150
  4. [4] A. Dalalyan. Sharp adaptive estimation of the drift function for ergodic diffusions. Ann. Statist. 33 (2005) 2507–2528. Zbl1084.62079MR2253093
  5. [5] A. Dalalyan and Y. Kutoyants. On second order minimax estimation of the invariant density for ergodic diffusions. Statist. Decisions 22 (2004) 17–41. Zbl1057.62066MR2065989
  6. [6] S. Delattre and M. Hoffmann. Asymptotic equivalence for a null recurrent diffusion model. Bernoulli 8 (2002) 139–174. Zbl1040.60067MR1895888
  7. [7] S. Delattre, M. Hoffmann and M. Kessler. Dynamics adaptive estimation of a scalar diffusion. Prépublication PMA-762, Univ. Paris 6. Available at www.proba.jussieu.fr/mathdoc/preprints/. 
  8. [8] L. Galtchouk and S. Pergamentchikov. Sequential nonparametric adaptive estimation of the drift coefficient in diffusion processes. Math. Methods Statist. 10 (2001) 316–330. Zbl1005.62070MR1867163
  9. [9] R. Höpfner and Y. Kutoyants. On a problem of statistical inference in null recurrent diffusions. Stat. Inference Stoch. Process. 6 (2003) 25–42. Zbl1012.62091MR1965183
  10. [10] R. Höpfner and E. Löcherbach. Limit Theorems for Null Recurrent Markov Processes. Providence, RI, 2003. Zbl1018.60074MR1949295
  11. [11] K. Itô and H. P. McKean, Jr.Diffusion Processes and Their Sample Paths. Springer, Berlin, 1974. Zbl0285.60063MR345224
  12. [12] R. Khasminskii. Limit distributions of some integral functionals for null-recurrent diffusions. Stochastic Process. Appl. 92 (2001) 1–9. Zbl1047.60078MR1815176
  13. [13] K. Kuratowski. Introduction a la theorie des ensembles et a la topologie. Institut de Mathematiques, Universite Geneve, 1966. Zbl0136.19002MR231338
  14. [14] Y. Kutoyants. Statistical Inference for Ergodic Diffusion Processes. Springer, London, 2004. Zbl1038.62073MR2144185
  15. [15] E. Löcherbach and D. Loukianova. On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multidimensional diffusions. To appear in Stochastic Process. Appl. Zbl1202.60122
  16. [16] D. Loukianova and O. Loukianov. Deterministic equivalents of additive functionals of recurrent diffusions and drift estimation. To appear in Stat. Inference Stoch. Process. Zbl1204.60032
  17. [17] D. Loukianova and O. Loukianov. Almost sure rate of convergence of maximum likelihood estimators for multidimensional diffusions. In Dependence in Probability and Statistics 269–347. Springer, New York, 2006. Zbl1102.62086MR2283262
  18. [18] Y. Nishiyama. A maximum inequality for continuous martingales and M-estimation in Gaussian white noise model. Ann. Statist. 27 (1999) 675–696. Zbl0954.62100MR1714712
  19. [19] D. Revuz and M. Yor. Continuous Martingales and Brownian Motion. Springer, Berlin, 1994. Zbl0804.60001MR1303781
  20. [20] L. C. G. Rogers and D. Williams. Diffusions, Markov Processes, and Martingales, Vol. 2, Wiley, New York, 1990. Zbl0826.60002MR921238
  21. [21] A. Touati. Théorèmes limites pour les processus de Markov récurrents. Unpublished paper, 1988. (See also C.R.A.S. Paris Série I 305 (1987) 841–844.) Zbl0627.60069MR923211
  22. [22] H. van Zanten. On empirical processes for ergodic diffusions and rates of convergence of M-estimators. Scand. J. Statist. 30 (2003) 443–458. Zbl1034.62071MR2002221
  23. [23] H. van Zanten. On the rate of convergence of the maximum likelihood estimator in Brownian semimartingale models. Bernoulli 11 (2005) 643–664. Zbl1092.62079MR2158254
  24. [24] N. Yoshida. Asymptotic behavior of M-estimators and related random field for diffusion process. Ann. Inst. Statist. Math. 42 (1990) 221–251. Zbl0723.62048MR1064786

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.