×

Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. (English) Zbl 0932.62097

Summary: Hidden Markov models (HMMs) have during the last decade become a widespread tool for modeling sequences of dependent random variables. Inference for such models is usually based on the maximum-likelihood estimator (MLE), and consistency of the MLE for general HMMs was recently proved by {B. G. Leroux} [Stochastic Processes Appl. 40, No. 1, 127-143 (1992; Zbl 0738.62081)]. We show that under mild conditions the MLE is also asymptotically normal and prove that the observed information matrix is a consistent estimator of the Fisher information.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
62M09 Non-Markovian processes: estimation

Citations:

Zbl 0738.62081
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Albert, P. S. (1991). A two-state Markov mixture model for a time series of epileptic seizure counts. Biometrics 47 1371-1381.
[2] Baum, L. E. and Petrie, T. (1966). Statistical inference for probabilistic functions of finite state Markov chains. Ann. Math. Statist. 37 1554-1563. · Zbl 0144.40902
[3] Baum, L. E., Petrie, T., Soules, G. and Weiss, N. (1970). A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statist. 41 164-171. · Zbl 0188.49603
[4] Bickel, P. J. and Ritov, Y. (1996). Inference in hidden Markov models I: local asy mptotic normality in the stationary case. Bernoulli 2 199-228. · Zbl 1066.62535
[5] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA. · Zbl 0709.60002
[6] Fredkin, D. R. and Rice, J. A. (1992). Maximum likelihood estimation and identification directly from single-channel recordings. Proc. Roy al Soc. London Ser. B 249 125-132.
[7] Guttorp, P. (1995). Stochastic Modeling of Scientific Data. Chapman & Hall, London. · Zbl 0862.60034
[8] Heffes, H. and Lucantoni, D. (1986). A Markov modulated characterization of packetized voice and data traffic and related statistical multiplexer performance. IEEE J. Select. Areas Comm. 4 856-867.
[9] Jamshidian, M. and Jennrich, R. I. (1997). Acceleration of the EM algorithm by using quasiNewton methods. J. Roy al Statist. Soc. Ser. B 59 569-587. JSTOR: · Zbl 0889.62042
[10] Le, N. D., Leroux, B. G. and Puterman, M. L. (1992). Reader reaction: exact likelihood evaluation in a Markov mixture model for time series of seizure counts. Biometrics 48 317-323.
[11] Leroux, B. G. (1992). Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 127-143. · Zbl 0738.62081
[12] Leroux, B. G. and Puterman, M. L. (1992). Maximum-penalized-likelihood estimation for independent and Markov-dependent mixture models. Biometrics 48 545-558.
[13] Lindgren, G. (1978). Markov regime models for mixed distributions and switching regressions. Scand. J. Statist. 5 81-91. · Zbl 0382.62073
[14] Louis, T. A. (1982). Finding the observed information matrix when using the EM algorithm. J. Roy al Statist. Soc. Ser. B 44 226-233. JSTOR: · Zbl 0488.62018
[15] MacDonald, I. L. and Zucchini, W. (1997). Hidden Markov and Other Models for Discrete-valued Time Series. Chapman & Hall, London. · Zbl 0868.60036
[16] McLachlan, G. J. and Krishnan, T. (1997). The EM Algorithm and Extensions. Wiley, New York. · Zbl 0882.62012
[17] Meng, X.-L. and van Dy k, D. (1997). The EM algorithm-an old folk-song sung to a new fast tune (with discussion). J. Roy al Statist. Soc. Ser. B 59 511-567. JSTOR: · Zbl 1090.62518
[18] Petrie, T. (1969). Probabilistic functions of finite state Markov chains. Ann. Math. Statist. 40 97-115. · Zbl 0181.21201
[19] Press, W. H., Flannery, B. P., Teukolsky, S. A. and Vetterling, W. T. (1989). Numerical Recipes. Cambridge Univ. Press. · Zbl 0698.65001
[20] Rabiner, L. R. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77 257-284.
[21] Ritov, Y. (1996). Uniform convergence of quasi-convex functions with applications to missing data and hidden Markov models.
[22] Ry dén, T. (1994). Parameter estimation for Markov modulated Poisson processes. Stochastic Models 10 795-829. · Zbl 0815.62059
[23] Shiry ayev, A. N. (1984). Probability. Springer, New York. · Zbl 0536.60001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.