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A recursive nonparametric estimator for the transition kernelof a piecewise-deterministic Markov process∗

Published online by Cambridge University Press:  22 October 2014

Romain Azaïs
Romain Azaïs*
Affiliation:
INRIA Bordeaux Sud-Ouest, team CQFD, France and Université Bordeaux, IMB, CNRS UMR 5251, 200 Avenue de la Vieille Tour, 33405 Talence cedex, France. romain.azais@inria.fr
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Abstract

In this paper, we investigate a nonparametric approach to provide a recursive estimatorof the transition density of a piecewise-deterministic Markov process, from only oneobservation of the path within a long time. In this framework, we do not observe a Markovchain with transition kernel of interest. Fortunately, one may write the transitiondensity of interest as the ratio of the invariant distributions of two embedded chains ofthe process. Our method consists in estimating these invariant measures. We state a resultof consistency and a central limit theorem under some general assumptions about the mainfeatures of the process. A simulation study illustrates the well asymptotic behavior ofour estimator.

Keywords

Piecewise-deterministic Markov processesnonparametric estimationrecursive estimatortransition kernelasymptotic consistency

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Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 726 - 749
Copyright
© EDP Sciences, SMAI 2014

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