Exact MLE and asymptotic properties for nonparametric semi-Markov models

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.

Keywords:

• asymptotic efficiency
• asymptotic normality
• backward recurrence times
• consistency
• maximum-likelihood estimation
• nonparametric estimation
• semi-Markov chains

AMS Subject Classification:

• 62F12
• 60K15
• 62M09

Acknowledgements

The authors are grateful to the editor, and two anonymous referees (including the associate editor). Their valuable comments and suggestions improved considerably this paper.