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Consistency of the maximum likelihood estimator for general hidden Markov models

Douc, Randal; Moulines, Eric; Olsson, Jimmy LU and van Handel, Ramon (2011) In Annals of Statistics 39(1). p.474-513
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Hidden Markov models, maximum likelihood estimation, strong, consistency, V-uniform ergodicity, concentration inequalities, state, space models
in
Annals of Statistics
volume
39
issue
1
pages
474 - 513
publisher
Inst Mathematical Statistics
external identifiers
  • wos:000288183800016
ISSN
0090-5364
DOI
10.1214/10-AOS834
language
English
LU publication?
yes
id
a3f8d624-ac24-4b20-9d0b-981f37756a5c (old id 1868749)
date added to LUP
2011-04-19 11:56:59
date last changed
2016-04-15 20:55:20
@article{a3f8d624-ac24-4b20-9d0b-981f37756a5c,
  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.},
  author       = {Douc, Randal and Moulines, Eric and Olsson, Jimmy and van Handel, Ramon},
  issn         = {0090-5364},
  keyword      = {Hidden Markov models,maximum likelihood estimation,strong,consistency,V-uniform ergodicity,concentration inequalities,state,space models},
  language     = {eng},
  number       = {1},
  pages        = {474--513},
  publisher    = {Inst Mathematical Statistics},
  series       = {Annals of Statistics},
  title        = {Consistency of the maximum likelihood estimator for general hidden Markov models},
  url          = {http://dx.doi.org/10.1214/10-AOS834},
  volume       = {39},
  year         = {2011},
}