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.
Mots-clés : piecewise-deterministic Markov processes, nonparametric estimation, recursive estimator, transition kernel, asymptotic consistency
@article{PS_2014__18__726_0,
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://archive.numdam.org/articles/10.1051/ps/2013054/}
}
TY - JOUR AU - Azaïs, Romain TI - A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process JO - ESAIM: Probability and Statistics PY - 2014 SP - 726 EP - 749 VL - 18 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ps/2013054/ DO - 10.1051/ps/2013054 LA - en ID - PS_2014__18__726_0 ER -
%0 Journal Article %A Azaïs, Romain %T A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process %J ESAIM: Probability and Statistics %D 2014 %P 726-749 %V 18 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ps/2013054/ %R 10.1051/ps/2013054 %G en %F PS_2014__18__726_0
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. http://archive.numdam.org/articles/10.1051/ps/2013054/
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