Abstract
The paper shows that the distribution of the normalized minimum contrast estimator of the drift parameter in the Ornstein–Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform error rate of the order O (T − 1/2). A precise estimate of the constant in the upper bound is also given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bishwal JPN (2000a) Sharp Berry–Esseen bound for the maximum likelihood estimator in the Ornstein–Uhlenbeck process. Sankhyā, Ser A 62(1):1–10
Bishwal JPN (2000b) Rates of convergence of the posterior distributions and the Bayes estimators in the Ornstein–Uhlenbeck process. Random Oper Stoch Equ 8(1):51–70
Bishwal JPN (2001) Accuracy of normal approximation for the maximum likelihood and Bayes estimators in the Ornstein–Uhlenbeck process using random normings. Stat Probab Lett 52(4):427–439
Bishwal JPN (2006) Rate of weak convergence of the approximate minimum contrast estimators for discretely observed Ornstein–Uhlenbeck process. Stat Probab Lett 76(13):1397–1409
Bishwal JPN (2008) Parameter estimation in stochastic differential equations. Lecture notes in mathematics, vol 1923. Springer
Bishwal JPN, Bose A (1995) Speed of convergence of the maximum likelihood estimator in the Ornstein–Uhlenbeck process. Calcutta Stat Assoc Bull 45:245–251
Bishwal JPN, Bose A (2001) Rates of convergence of approximate maximum likelihood estimators in the Ornstein–Uhlenbeck process. Comput Math Appl 42(1–2):23–38
Borokov AA (1973) On the rate of convergence for the invariance principle. Theory Probab Appl 18:217–234
Feller W (1957) An introduction to probability theory and its applications, vol I. Wiley, New York
Friedman A (1975) Stochastic differential equations, vol I. Academic, New York
Florens-Landais D, Pham H (1999) Large deviations in estimation of an Ornstein–Uhlenbeck model. J Appl Probab 36:60–77
Kutoyants Yu A (1994) Identification of dynamical systems with small noise. Kluwer, Dordrecht
Lanksa V (1979) Minimum contrast estimation in diffusion processes. J Appl Probab 16:65–75
Michel R (1973) The bound in the Berry–Esseen result for minimum contrast estimates. Metrika 20:148–155
Mishra MN, Prakasa Rao BLS (1985) On the Berry–Esseen theorem for maximum likelihood estimator for linear homogeneous diffusion processes. Sankhyā, Ser A 47:392–398
Pfanzagl J (1971) The Berry–Esseen bound for minimum contrast estimators. Metrika 17:82–91
Prakasa Rao BLS (1999) Statistical inference for diffusion type processes. Arnold, London
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bishwal, J.P.N. Uniform Rate of Weak Convergence of the Minimum Contrast Estimator in the Ornstein–Uhlenbeck Process. Methodol Comput Appl Probab 12, 323–334 (2010). https://doi.org/10.1007/s11009-008-9099-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-008-9099-x
Keywords
- Itô stochastic differential equation
- Ornstein–Uhlenbeck process
- Minimum contrast estimator
- Uniform rate of weak convergence
- Fourier method