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A recursive nonparametric estimator for the transition kernel of 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 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.

Keywords

Piecewise-deterministic Markov processesnonparametric estimationrecursive estimatortransition kernelasymptotic consistency
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 726 - 749
Copyright
© EDP Sciences, SMAI 2014

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References

T. Aven and U. Jensen, Stochastic models in reliability, vol. 41 of Applications of Mathematics. Springer-Verlag, New York (1999).
R. Azaïs, F. Dufour and A. Gégout-Petit, Nonparametric estimation of the conditional density of the inter-jumping times for piecewise-deterministic Markov processes. Preprint arXiv:1202.2212v2 (2012).
Basu, A.K. and Sahoo, D.K., On Berry-Esseen theorem for nonparametric density estimation in Markov sequences. Bull. Inform. Cybernet. 30 (1998) 25–39. Google Scholar
Chafaï, D., Malrieu, F. and Paroux, K., On the long time behavior of the TCP window size process. Stoch. Process. Appl. 120 (2010) 1518–1534. Google Scholar
Chiquet, J. and Limnios, N., A method to compute the transition function of a piecewise deterministic Markov process with application to reliability. Statist. Probab. Lett. 78 (2008) 1397–1403. Google Scholar
Clémençon, S.J.M., Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist. 9 (2000) 323–357. Google Scholar
M.H.A. Davis, Markov models and optimization, vol. 49 of Monogr. Statist. Appl. Probab. Chapman & Hall, London (1993).
Doukhan, P. and Ghindès, M., Estimation de la transition de probabilité d’une chaîne de Markov Doëblin-récurrente. Étude du cas du processus autorégressif général d’ordre 1. Stoch. Process. Appl. 15 (1983) 271–293. Google Scholar
M. Doumic, M. Hoffmann, N. Krell and L. Robert, Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Preprint (2012).
M. Duflo, Random iterative models. Appl. Math. Springer-Verlag, Berlin (1997).
Genadot, A. and Thieullen, M., Averaging for a fully coupled Piecewise Deterministic Markov Process in Infinite Dimensions. Adv. Appl. Probab. 44 3 (2012). Google Scholar
Hernández-Lerma, O., Esparza, S.O. and Duran, B.S., Recursive nonparametric estimation of nonstationary Markov processes. Bol. Soc. Mat. Mexicana 33 (1988) 57–69. Google Scholar
O. Hernández-Lerma and J.B. Lasserre, Markov chains and invariant probabilities, vol. 211 of Progr. Math. Birkhäuser Verlag, Basel (2003).
J. Hu, W. Wu and S. Sastry, Modeling subtilin production in bacillus subtilis using stochastic hybrid systems. Hybrid Systems: Computation and Control. Edited by R. Alur and G.J. Pappas. Lect. Notes Comput. Sci. Springer-Verlag, Berlin (2004).
Lacour, C., Adaptive estimation of the transition density of a Markov chain. Ann. Inst. Henri Poincaré, Probab. Statist. 43 (2007) 571–597. Google Scholar
Lacour, C., Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stoch. Process. Appl. 118 (2008) 232–260. Google Scholar
Liebscher, E., Density estimation for Markov chains. Statistics 23 (1992) 27–48. Google Scholar
Masry, E. and Györfi, L., Strong consistency and rates for recursive probability density estimators of stationary processes. J. Multivariate Anal. 22 (1987) 79–93. Google Scholar
S. Meyn and R.L. Tweedie, Markov chains and stochastic stability, second edition. Cambridge University Press, Cambridge (2009).
M. Rosenblatt, Density estimates and Markov sequences. In Nonparametric Techniques in Statistical Inference. Proc. of Sympos., Indiana Univ., Bloomington, Ind., 1969. Cambridge Univ. Press, London (1970), 199–213.
Roussas, G.G., Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 (1969) 73–87. Google Scholar
Van Ryzin, J., On strong consistency of density estimates. Ann. Math. Statist. 40 (1969) 1765–1772. Google Scholar
Yakowitz, S., Nonparametric density and regression estimation for Markov sequences without mixing assumptions. J. Multivariate Anal. 30 (1989) 124–136. Google Scholar