Hidden Markov model for parameter estimation of a random walk in a Markov environment

¹ Laboratoire MAPMO, UMR CNRS 6628, Fédération Denis-Poisson, Université d’Orléans, Orléans, France
Pierre.Andreoletti@univ-orleans.fr
² Laboratoire de Mathématiques et Modélisation d’Évry, Université d’Évry Val d’Essonne, UMR CNRS 8071, Évry, France
dasha.loukianova@maths.univ-evry.fr
³ Laboratoire de Probabilités et Modèles Aléatoires, UMR CNRS 7599, Université Pierre et Marie Curie, Université Paris Diderot, Paris, France
catherine.matias@math.cnrs.fr

Received: 19 November 2014
Revised: 1 April 2015

Abstract

We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.