Hostname: page-component-8448b6f56d-c4f8m Total loading time: 0 Render date: 2024-04-16T12:07:01.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Large deviations in estimation of an Ornstein-Uhlenbeck model

Part of: Limit theorems

Published online by Cambridge University Press:  14 July 2016

Danielle Florens-Landais  and
Huyên Pham
Danielle Florens-Landais*
Affiliation:
CEREMADE
Huyên Pham*
Affiliation:
Université de Marne-la-Vallée
*
∗Postal address: CEREMADE, 42 Rue de la Procession, 75015 Paris, France.
∗∗Postal address: Equipe d'Analyse et de Mathématiques Appliquées, Université de Marne la Vallée, Cité Descartes, 5 Boulevard Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée, Cedex 2, France. Email address: pham@math.univ-mlv.fr.
Get access Rights & Permissions [Opens in a new window]

Abstract

A large deviation principle (LDP) with an explicit rate function is proved for the estimation of drift parameter of the Ornstein-Uhlenbeck process. We establish an LDP for two estimating functions, one of them being the score function. The first one is derived by applying the Gärtner–Ellis theorem. But this theorem is not suitable for the LDP on the score function and we circumvent this key point by using a parameter-dependent change of measure. We then state large deviation principles for the maximum likelihood estimator and another consistent drift estimator.

Keywords

Large deviationsrate functionOrnstein–Uhlenbeck diffusion processdrift estimation

MSC classification

Secondary: 60F10: Large deviations
Type
Research Papers
Information
Journal of Applied Probability , Volume 36 , Issue 1 , March 1999 , pp. 60 - 77
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bahadur, R., Gupta, J. C., and Zabell, S. (1980). Large deviations, tests and estimates. In Asymptotic Theory of Statistical tests and Estimation, ed. Chakravarti, I. M. Academic press, New York, pp. 33–64.Google Scholar
Bercu, B., Gamboa, F., and Rouault, A. (1997). Large deviations for quadratic forms of gaussian stationary processes. Stoch. Proc. Appl. 71, 75–90.CrossRefGoogle Scholar
Bryc, W., and Smolenski, W. (1993). On the large deviation principle for a quadratic functional of the autoregressive process. Statist. Prob. Lett. 17, 281–285.CrossRefGoogle Scholar
Bryc, W., and Dembo, A. (1997). Large deviations for quadratic functionals of gaussian processes. J. Theoretical Prob. 10, 307–332.CrossRefGoogle Scholar
Dembo, A., and Zeitouni, O. (1993). Large deviations techniques and applications. Bartlett, Boston, MA.Google Scholar
Florens, D. (1984). Un théorème de limite centrale pour une diffusion et sa discrétisée. C. R. Acad. Sci. Paris, t. 299, série I, 19, 995–998.Google Scholar
Hoffmann-Jorgensen, J. (1994). Probability with a View Towards Statistics, Vol. 1. Chapman and Hall, New York.CrossRefGoogle Scholar