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Exact MLE and asymptotic properties for nonparametric semi-Markov models

Samis Trevezas 1, 2 Nikolaos Limnios 3 
2 DIGIPLANTE - Modélisation de la croissance et de l'architecture des plantes
MAS - Mathématiques Appliquées aux Systèmes - EA 4037, Cirad - Centre de Coopération Internationale en Recherche Agronomique pour le Développement, Inria Saclay - Ile de France
Abstract : This article concerns maximum-likelihood estimation for discrete time homogeneous nonparametric semi-Markov models with finite state space. In particular, we present the exact maximum-likelihood estimator of the semi-Markov kernel which governs the evolution of the semi-Markov chain (SMC). We study its asymptotic properties in the following cases: (i) for one observed trajectory, when the length of the observation tends to infinity, and (ii) for parallel observations of independent copies of an SMC censored at a fixed time, when the number of copies tends to infinity. In both cases, we obtain strong consistency, asymptotic normality, and asymptotic efficiency for every finite dimensional vector of this estimator. Finally, we obtain explicit forms for the covariance matrices of the asymptotic distributions.
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Samis Trevezas, Nikolaos Limnios. Exact MLE and asymptotic properties for nonparametric semi-Markov models. Journal of Nonparametric Statistics, American Statistical Association, 2011, 23 (3), pp.719-739. ⟨10.1080/10485252.2011.555543⟩. ⟨inria-00625293⟩

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