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.
Classification : 62G05, 62M05
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://www.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 DA - 2014/// SP - 726 EP - 749 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2013054/ UR - https://doi.org/10.1051/ps/2013054 DO - 10.1051/ps/2013054 LA - en ID - PS_2014__18__726_0 ER -
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://www.numdam.org/articles/10.1051/ps/2013054/
[1] and , Stochastic models in reliability, vol. 41 of Applications of Mathematics. Springer-Verlag, New York (1999). | MR 1679540 | Zbl 0926.60006
[2] , and , Nonparametric estimation of the conditional density of the inter-jumping times for piecewise-deterministic Markov processes. Preprint arXiv:1202.2212v2 (2012).
[3] and , On Berry-Esseen theorem for nonparametric density estimation in Markov sequences. Bull. Inform. Cybernet. 30 (1998) 25-39. | MR 1629735 | Zbl 0921.62039
[4] , and , On the long time behavior of the TCP window size process. Stoch. Process. Appl. 120 (2010) 1518-1534. | MR 2653264 | Zbl 1196.68028
[5] and , A method to compute the transition function of a piecewise deterministic Markov process with application to reliability. Statist. Probab. Lett. 78 (2008) 1397-1403. | MR 2528336 | Zbl 1190.60065
[6] , Adaptive estimation of the transition density of a regular Markov chain. Math. Methods Statist. 9 (2000) 323-357. | MR 1827473 | Zbl 1008.62076
[7] , Markov models and optimization, vol. 49 of Monogr. Statist. Appl. Probab. Chapman & Hall, London (1993). | MR 1283589 | Zbl 0780.60002
[8] and , 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. | MR 711186 | Zbl 0515.62037
[9] , , and , Statistical estimation of a growth-fragmentation model observed on a genealogical tree. Preprint (2012).
[10] , Random iterative models. Appl. Math. Springer-Verlag, Berlin (1997). | MR 1485774 | Zbl 0868.62069
[11] and , Averaging for a fully coupled Piecewise Deterministic Markov Process in Infinite Dimensions. Adv. Appl. Probab. 44 3 (2012). | MR 3024608 | Zbl 1276.60023
[12] , and , Recursive nonparametric estimation of nonstationary Markov processes. Bol. Soc. Mat. Mexicana 33 (1988) 57-69. | MR 1110000 | Zbl 0732.62086
[13] and , Markov chains and invariant probabilities, vol. 211 of Progr. Math. Birkhäuser Verlag, Basel (2003). | MR 1974383 | Zbl 1036.60003
[14] , and , 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). | Zbl 1135.93307
[15] , Adaptive estimation of the transition density of a Markov chain. Ann. Inst. Henri Poincaré, Probab. Statist. 43 (2007) 571-597. | Numdam | MR 2347097 | Zbl 1125.62087
[16] , Nonparametric estimation of the stationary density and the transition density of a Markov chain. Stoch. Process. Appl. 118 (2008) 232-260. | MR 2376901 | Zbl 1129.62028
[17] , Density estimation for Markov chains. Statistics 23 (1992) 27-48. | MR 1245707 | Zbl 0809.62077
[18] and , Strong consistency and rates for recursive probability density estimators of stationary processes. J. Multivariate Anal. 22 (1987) 79-93. | MR 890884 | Zbl 0619.62079
[19] and , Markov chains and stochastic stability, second edition. Cambridge University Press, Cambridge (2009). | MR 2509253 | Zbl 1165.60001
[20] , 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. | MR 278468
[21] , Nonparametric estimation in Markov processes. Ann. Inst. Statist. Math. 21 (1969) 73-87. | MR 247722 | Zbl 0181.45804
[22] , On strong consistency of density estimates. Ann. Math. Statist. 40 (1969) 1765-1772. | MR 258172 | Zbl 0198.23502
[23] , Nonparametric density and regression estimation for Markov sequences without mixing assumptions. J. Multivariate Anal. 30 (1989) 124-136. | MR 1003712 | Zbl 0701.62049
Cité par Sources :
