The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains

Randal Douc (1) , François Roueff (2) , Tepmony Sim (2)

1 TSP - CITI - Communications, Images et Traitement de l'Information
2 LTCI - Laboratoire Traitement et Communication de l'Information

This paper deals with a parametrized family of partially observed bivariate Markov chains. We establish, under very mild assumptions, that the limit of the normalized log-likelihood is maximized when the parameter belongs to the equivalence class of the true parameter, which is a key feature for obtaining the consistency of the Maximum Likelihood Estimators (MLE) in well-specified models. This result is obtained in the general framework of partially dominated models. We examine two specific cases of interest, namely, Hidden Markov models and Observation-Driven times series. In contrast with previous approaches, the identifiability is addressed by relying on the unicity of the invariant distribution of the Markov chain associated to the complete data, regardless its rate of convergence to the equilibrium.

Mots clés: time series of counts; observation-driven models; maximum likelihood; hidden Markov models; ergodicity; consistency

Domaines: Statistiques [stat]

Randal Douc, François Roueff, Tepmony Sim. The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains. 2014. ⟨hal-01080955v1⟩