Volume 18, 2014
|Page(s)||726 - 749|
|Published online||22 October 2014|
A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process∗
INRIA Bordeaux Sud-Ouest, team CQFD, France and Université Bordeaux, IMB,
CNRS UMR 5251, 200 Avenue de la Vieille Tour, 33405 Talence cedex, France
Revised: 2 August 2013
In this paper, we investigate a nonparametric approach to provide a recursive estimator of the transition density of a piecewise-deterministic Markov process, from only one observation of the path within a long time. In this framework, we do not observe a Markov chain with transition kernel of interest. Fortunately, one may write the transition density of interest as the ratio of the invariant distributions of two embedded chains of the process. Our method consists in estimating these invariant measures. We state a result of consistency and a central limit theorem under some general assumptions about the main features of the process. A simulation study illustrates the well asymptotic behavior of our estimator.
Mathematics Subject Classification: 62G05 / 62M05
Key words: Piecewise-deterministic Markov processes / nonparametric estimation / recursive estimator / transition kernel / asymptotic consistency
© EDP Sciences, SMAI 2014
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