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February 2011 Consistency of the maximum likelihood estimator for general hidden Markov models
Randal Douc, Eric Moulines, Jimmy Olsson, Ramon van Handel
Ann. Statist. 39(1): 474-513 (February 2011). DOI: 10.1214/10-AOS834


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.


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Randal Douc. Eric Moulines. Jimmy Olsson. Ramon van Handel. "Consistency of the maximum likelihood estimator for general hidden Markov models." Ann. Statist. 39 (1) 474 - 513, February 2011.


Published: February 2011
First available in Project Euclid: 15 February 2011

zbMATH: 1209.62194
MathSciNet: MR2797854
Digital Object Identifier: 10.1214/10-AOS834

Primary: 60F10 , 62B10 , 62F12 , 62M09
Secondary: 60J05 , 62M05 , 62M10 , 94A17

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

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 1 • February 2011
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