Consistency of the Maximum Likelihood Estimator for general hidden Markov models

Randal Douc (1) , Eric Moulines (2) , Jimmy Olsson (3) , Ramon van Handel (4)

1 TSP - CITI - Communications, Images et Traitement de l'Information
2 LTCI - Laboratoire Traitement et Communication de l'Information
3 CMS - Center for Mathematical Sciences
4 Operations Research & Financial Engineering

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.

Keywords: general state space; time series; measures of information; entropy; large deviations; information theory in statistics; parametric inference: asymptotics of estimators; non-Markov processes estimation; discrete time Markov processes

Domains: Mathematics [math]; Statistics [math.ST]; Statistics [stat]; Statistics Theory [stat.TH]

Randal Douc, Eric Moulines, Jimmy Olsson, Ramon van Handel. Consistency of the Maximum Likelihood Estimator for general hidden Markov models. 2009. ⟨hal-00442774⟩