Geodesic
A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process
ESAIM: Probability and Statistics, Tome 18 (2014), pp. 726-749

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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.

DOI : 10.1051/ps/2013054
Classification : 62G05, 62M05
Keywords: piecewise-deterministic Markov processes, nonparametric estimation, recursive estimator, transition kernel, asymptotic consistency 
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     author = {Aza{\"\i}s, Romain},
     title = {A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic {Markov} process},
     journal = {ESAIM: Probability and Statistics},
     pages = {726--749},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013054},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/ps/2013054/}
}
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Azaïs, Romain. A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 726-749. doi: 10.1051/ps/2013054

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