Abstract
Consider non-recurrent Ornstein–Uhlenbeck processes with unknown drift and diffusion parameters. Our purpose is to estimate the parameters jointly from discrete observations with a certain asymptotics. We show that the likelihood ratio of the discrete samples has the uniform LAMN property, and that some kind of approximated MLE is asymptotically optimal in a sense of asymptotic maximum concentration probability. The estimator is also asymptotically efficient in ergodic cases.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aldous D.J., Eagleson G.K. (1978) On mixing and stability of limit theorems. The Annals Probability 6: 325–333
Anderson T.W. (1959) On asymptotic distributions of estimates of parameters of stochastic difference equations. The Annals of Mathematical Statistics 30: 676–687
Basawa I.V., Scott D.J. (1983) Asymptotic optimal inference for non-ergodic models. Springer, New York
Dietz H.M., Kutoyants A.Yu. (2003) Parameter estimation for some non-recurrent solutions of SDE. Statistics & Decisions 21(1): 29–45
Feigin P.D. (1976) Maximum likelihood estimation for continuous-time stochastic processes. Advances in Applied Probability 8(4): 712–736
Feigin P.D. (1979) Some comments concerning a curious singularity. Journal of Applied Probability 16(2): 440–444
Gobet E. (2002) LAN property for ergodic diffusions with discrete observations. Annales de l’Institut Henri Poincaré Probabilités et Statistiques 38(5): 711–737
Hall P., Heyde C. (1980) Martingale limit theory and its applications. Academic Press, New York
Ibragimov I.A., Has’minskii R.Z. (1981) Statistical estimation. Springer, Berlin
Jacod J. (2006) Parametric inference for discretely observed non-ergodic diffusions. Bernoulli 12(3): 383–401
Jeganathan P. (1982) On the asymptotic theory of estimation when the limit of the loglikelihood ratio is mixed normal. Sankhya Series A 44: 173–212
Kasonga R.A. (1988) The consistency of a nonlinear least squares estimator from diffusion processes. Stochastic Processes and their Applications 30: 263–275
Kasonga R.A. (1990) Parameter estimation by deterministic approximation of a solution of a stochastic differential equation. Communications in Statistics. Stochastic Models 6: 59–67
Kessler M. (1997) Estimation of Diffusion Processes From Discrete Observations. Scandinavian Journal of Statistics 24: 211–229
Kutoyants Y.A. (2004) Statistical inference for ergodic diffusion processes. Springer, London
Le Breton, A. (1975). On continuous and discrete sampling for parameter estimation in diffusion type processes. Stochastic systems: Modeling, identification and optimization. In Proceedings of Symposium of University of Kentucky, Lexington, KY.
Le Breton, A. (1976). Mathematical programming study (Vol. 5, p. 124–144).
Luschgy H. (1992) Local asymptotic mixed normality for semimartingale experiments. Probability Theory and Related Fields 92: 151–176
Phillips P.C.B. (1987) Towards a unified asymptotic theory for autoregression. Biometrika 74(3): 535–547
Prakasa Rao B.L.S. (1999a) Semimartingales and their statistical inference, Monographs on Statistics and Applied Probability, 83. Boca Raton, Chapman & Hall/CRC
Prakasa Rao B.L.S. (1999b) Statistical inference for diffusion type processes, Kendall’s Library of statistics (Vol. 8). Edward Arnold/Oxford University Press, London/New York
Shimizu, Y. (2009a). Quadratic type contrast functions for discretely observed non-ergodic diffusion processes (Research Report Series, 09-04). Department of Engineering Science, Osaka University.
Shimizu Y. (2009) Notes on drift estimation for non-recurrent Ornstein–Uhlenbeck processes from sampled data. Statistics & Probability Letters 79(20): 2200–2207
Weiss L., Wolfowitz J. (1974) Maximum probability estimators and related topics. Springer Lecture Notes in Mathematics (Vol. 424). Springer, Berlin
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Shimizu, Y. Local asymptotic mixed normality for discretely observed non-recurrent Ornstein–Uhlenbeck processes. Ann Inst Stat Math 64, 193–211 (2012). https://doi.org/10.1007/s10463-010-0307-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-010-0307-4