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

Randal Douc (1, 2, 3) , François Roueff (4) , Tepmony Sim (4)

1 TIPIC-SAMOVAR - Traitement de l'Information Pour Images et Communications
2 TSP - CITI - Communications, Images et Traitement de l'Information
3 CNRS - Centre National de la Recherche Scientifique
4 LTCI - Laboratoire Traitement et Communication de l'Information

This paper deals with a parametrized family of partially observed bivariate Markov chains. We establish that, under very mild assumptions, the limit of the normalized log-likelihood function is maximized when the parameters belong to the equivalence class of the true parameter, which is a key feature for obtaining the consistency of the maximum likelihood estimators (MLEs) 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 (HMMs) and observation-driven time series models. In contrast with previous approaches, the identifiability is addressed by relying on the uniqueness of the invariant distribution of the Markov chain associated to the complete data, regardless its rate of convergence to the equilibrium.

Mots clés: Maximum likelihood; Observation-driven models; Time series of counts; Hidden Markov models; Consistency; Ergodicity

Domaines: Théorie [stat.TH]; Statistiques [stat]

Randal Douc, François Roueff, Tepmony Sim. The maximizing set of the asymptotic normalized log-likelihood for partially observed Markov chains. The Annals of Applied Probability, 2016, 26 (4), pp.2357 - 2383. ⟨10.1214/15-AAP1149⟩. ⟨hal-01080955v2⟩