The Annals of Statistics

Asymptotic normality of the maximum likelihood estimator in state space models

Jens Ledet Jensen and Niels Væver Petersen

Full-text: Open access

Abstract

State space models is a very general class of time series models capable of modelling dependent observations in a natural and interpretable way. Inference in such models has been studied by Bickel, Ritov and Rydén, who consider hidden Markov models, which are special kinds of state space models, and prove that the maximum likelihood estimator is asymptotically normal under mild regularity conditions. In this paper we generalize the results of Bickel, Ritov and Rydén to state space models, where the latent process is a continuous state Markov chain satisfying regularity conditions, which are fulfilled if the latent process takes values in a compact space.

Article information

Source
Ann. Statist., Volume 27, Number 2 (1999), 514-535.

Dates
First available in Project Euclid: 5 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1018031205

Digital Object Identifier
doi:10.1214/aos/1018031205

Mathematical Reviews number (MathSciNet)
MR1714719

Zentralblatt MATH identifier
0952.62023

Subjects
Primary: 62F12: Asymptotic properties of estimators
Secondary: 62M09: Non-Markovian processes: estimation

Keywords
State space models asymptotic normality maximum likelihood estimation.

Citation

Jensen, Jens Ledet; Petersen, Niels Væver. Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 (1999), no. 2, 514--535. doi:10.1214/aos/1018031205. https://projecteuclid.org/euclid.aos/1018031205


Export citation

References

  • Bickel, P. J., Ritov, Y. and Ryd´en, T. (1998). Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 1614-1635.
  • Doob, J. L. (1953). Stochastic Processes. Wiley, New York.
  • Durbin, J. and Koopman, S. J. (1997). Monte Carlo maximum likelihood estimation for nonGaussian state space models. Biometrika 84 669-684.
  • Fr ¨uhwirth-Schnatter, S. (1994). Applied state space modelling of non-Gaussian time series using integration-based Kalman filtering. Statist. Comput. 4 259-269.
  • Hoffmann-Jørgensen, J. (1994). Probability with a View toward Statistics 1. Chapman and Hall, London.
  • Jensen, J. L. (1986). Nogle asymptotiske resultater. Unpublished lecture notes. Univ. Aarhus.
  • Kitagawa, G. and Gersch, W. (1996). Smoothness Priors Analysis of Time Series. Springer, New York.
  • Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Process Appl. 40 127-143.
  • Shephard, N. and Pitt, M. K. (1997). Likelihood analysis of non-Gaussian measurement time series. Biometrika 84 653-667.
  • Sweeting, T. (1980). Uniform asymptotic normality of the maximum likelihood estimator. Ann. Statist. 8 1375-1381.
  • West, M. and Harrison, J. (1989). Bayesian Forecasting and Dynamic Models. Springer, New York.