The Annals of Statistics

Consistency of the maximum likelihood estimator for general hidden Markov models

Randal Douc, Eric Moulines, Jimmy Olsson, and Ramon van Handel

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Abstract

Consider a parametrized family of general hidden Markov models, where both the observed and unobserved components take values in a complete separable metric space. We prove that the maximum likelihood estimator (MLE) of the parameter is strongly consistent under a rather minimal set of assumptions. As special cases of our main result, we obtain consistency in a large class of nonlinear state space models, as well as general results on linear Gaussian state space models and finite state models.

A novel aspect of our approach is an information-theoretic technique for proving identifiability, which does not require an explicit representation for the relative entropy rate. Our method of proof could therefore form a foundation for the investigation of MLE consistency in more general dependent and non-Markovian time series. Also of independent interest is a general concentration inequality for V-uniformly ergodic Markov chains.

Article information

Source
Ann. Statist. Volume 39, Number 1 (2011), 474-513.

Dates
First available in Project Euclid: 15 February 2011

Permanent link to this document
http://projecteuclid.org/euclid.aos/1297779854

Digital Object Identifier
doi:10.1214/10-AOS834

Mathematical Reviews number (MathSciNet)
MR2797854

Zentralblatt MATH identifier
1209.62194

Subjects
Primary: 60F10: Large deviations 62B10: Information-theoretic topics [See also 94A17] 62F12: Asymptotic properties of estimators 62M09: Non-Markovian processes: estimation
Secondary: 60J05: Discrete-time Markov processes on general state spaces 62M05: Markov processes: estimation 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 94A17: Measures of information, entropy

Keywords
Hidden Markov models maximum likelihood estimation strong consistency V-uniform ergodicity concentration inequalities state space models

Citation

Douc, Randal; Moulines, Eric; Olsson, Jimmy; van Handel, Ramon. Consistency of the maximum likelihood estimator for general hidden Markov models. Ann. Statist. 39 (2011), no. 1, 474--513. doi:10.1214/10-AOS834. http://projecteuclid.org/euclid.aos/1297779854.


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