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Consistency of the maximum likelihood estimator for general hidden Markov models. (English) Zbl 1209.62194

Summary: 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.

MSC:

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
62B10 Statistical aspects of information-theoretic topics
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
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