Maximum likelihood estimation in partially observed Markov models with applications to time series of counts

Abstract : Maximum likelihood estimation is a widespread method for identifying a parametrized model of a time series from a sample of observations. Under the framework of well-specified models, it is of prime interest to obtain consistency of the estimator, that is, its convergence to the true parameter as the sample size of the observations goes to infinity. For many time series models, for instance hidden Markov models (HMMs), such a “strong” consistency property can however be difficult to establish. Alternatively, one can show that the maximum likelihood estimator (MLE) is consistent in a weakened sense, that is, as the sample size goes to infinity, the MLE eventually converges to a set of parameters, all of which associate to the same probability distribution of the observations as for the true one. The consistency in this sense, which remains a preferred property in many time series applications, is referred to as equivalence-class consistency. The task of deriving such a property generally involves two important steps: 1) show that the MLE converges to the maximizing set of the asymptotic normalized loglikelihood; and 2) show that any parameter in this maximizing set yields the same distribution of the observation process as for the true parameter. In this thesis, our primary attention is to establish the equivalence-class consistency for time series models that belong to the class of partially observed Markov models (PMMs) such as HMMs and observation-driven models (ODMs).
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Tepmony Sim. Maximum likelihood estimation in partially observed Markov models with applications to time series of counts. Statistics Theory [stat.TH]. Télécom ParisTech, 2016. English. ⟨NNT : 2016ENST0020⟩. ⟨tel-01458087⟩

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