Mathematics – Statistics Theory
Scientific paper
2009-12-22
Annals of Statistics 2011, Vol. 39, No. 1, 474-513
Mathematics
Statistics Theory
Published in at http://dx.doi.org/10.1214/10-AOS834 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Scientific paper
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.
Douc Randal
Handel Ramon van
Moulines Eric
Olsson Jimmy
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