no. 4
Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1204-1231.

Ce papier est consacré à l'estimation non-paramétrique du taux de saut et du taux de saut cumulé pour une classe générale de processus de renouvellement marqués non-homogènes, définis sur un espace métrique séparable. Dans notre cadre de travail, l'estimation nécessite seulement une observation du processus en temps long. Notre approche est basée sur une généralisation du modèle à intensité multiplicative introduit par Aalen dans les années soixante-dix. Nous donnons des estimateurs consistants de ces deux fonctions, sous des hypothèses portant sur l'ergodicité d'une chaîne immergée et sur les caractéristiques du processus. Le papier est illustré par un exemple numérique.

This paper is devoted to the nonparametric estimation of the jump rate and the cumulative rate for a general class of non-homogeneous marked renewal processes, defined on a separable metric space. In our framework, the estimation needs only one observation of the process within a long time. Our approach is based on a generalization of the multiplicative intensity model, introduced by Aalen in the seventies. We provide consistent estimators of these two functions, under some assumptions related to the ergodicity of an embedded chain and the characteristics of the process. The paper is illustrated by a numerical example.

DOI : 10.1214/12-AIHP503
Classification : 62G05, 62M09
Mots clés : non-homogeneous marked renewal process, nonparametric estimation, jump rate estimation, Nelson-Aalen estimator, asymptotic consistency, ergodicity of Markov chains 
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Azaïs, Romain; Dufour, François; Gégout-Petit, Anne. Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 4, pp. 1204-1231. doi : 10.1214/12-AIHP503. http://www.numdam.org/articles/10.1214/12-AIHP503/

[1] O. O. Aalen. Statistical inference for a family of counting processes. ProQuest LLC, Ann Arbor, MI. Ph.D. thesis, Univ. of California, Berkeley, 1975. | MR

[2] O. O. Aalen. Weak convergence of stochastic integrals related to counting processes. Z. Wahrsch. Verw. Gebiete 38 (1977) 261-277. | MR

[3] O. O. Aalen. Nonparametric inference for a family of counting processes. Ann. Statist. 6 (1978) 701-726. | MR

[4] O. O. Aalen, P. K. Andersen, Ø. Borgan, R. D. Gill and N. Keiding. History of applications of martingales in survival analysis. J. Électron. Hist. Probab. Stat. 5 (2009) 28. | MR

[5] P. K. Andersen, Ø. Borgan, R. D. Gill and N. Keiding. Statistical Models Based on Counting Processes. Springer, New York, 1993. | MR

[6] R. Azaïs, F. Dufour and A. Gégout-Petit. Nonparametric estimation of the conditional distribution of the inter-jumping times for piecewise-deterministic Markov processes. Preprint, 2012. Available at arXiv:1202.2212v2.

[7] M. Benaïm and N. El Karoui. Promenade aléatoire, Chaînes de Markov et simulations; martingales et stratégie. Les éd. de l'École polytechnique, Palaiseau, Paris, 2004.

[8] J. Beran. Nonparametric regression with randomly censored survival data. Technical report, Dept. Statist., Univ. California, Berkeley, 1981.

[9] K. L. Chung. A Course in Probability Theory, 2nd edition. Probability and Mathematical Statistics 21. Academic Press, New York, 1974. | MR

[10] F. Comte, S. Gaïffas and A. Guilloux. Adaptive estimation of the conditional intensity of marker-dependent counting processes. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011) 1171-1196. | MR

[11] D. R. Cox. Regression models and life-tables. J. Roy. Statist. Soc. Ser. B 34 (1972) 187-220. | MR

[12] D. M. Dabrowska. Nonparametric regression with censored survival time data. Scand. J. Statist. 14 (1987) 181-197. | MR

[13] M. H. A. Davis. Markov Models and Optimization. Monographs on Statistics and Applied Probability 49. Chapman & Hall, London, 1993.

[14] O. Hernández-Lerma and J. B. Lasserre. Further criteria for positive Harris recurrence of Markov chains. Proc. Amer. Math. Soc. 129 (2001) 1521-1524. | MR

[15] O. Hernández-Lerma and J. B. Lasserre. Markov Chains and Invariant Probabilities. Progress in Mathematics 211. Birkhäuser, Basel, 2003. | MR

[16] M. Jacobsen. Statistical Analysis of Counting Processes. Lecture Notes in Statistics 12. Springer, New York, 1982. | MR

[17] O. Kallenberg. Foundations of Modern Probability, 2nd edition. Probability and Its Applications (New York). Springer, New York, 2002. | MR

[18] A. Klenke. Probability Theory: A Comprehensive Course. Universitext. Springer, London, 2008. | MR

[19] G. Li and H. Doss. An approach to nonparametric regression for life history data using local linear fitting. Ann. Statist. 23 (1995) 787-823. | MR

[20] T. Martinussen and T. H. Scheike. Dynamic Regression Models for Survival Data. Statistics for Biology and Health. Springer, New York, 2006. | MR

[21] I. W. Mckeague and K. J. Utikal. Inference for a nonlinear counting process regression model. Ann. Statist. 18 (1990) 1172-1187. | MR

[22] S. Meyn and R. L. Tweedie. Markov Chains and Stochastic Stability, 2nd edition. Cambridge Univ. Press, Cambridge, 2009. | MR

[23] H. Ramlau-Hansen. Smoothing counting process intensities by means of kernel functions. Ann. Statist. 11 (1983) 453-466. | MR

[24] W. Stute. Conditional empirical processes. Ann. Statist. 14 (1986) 638-647. | MR

[25] K. J. Utikal. Nonparametric inference for a doubly stochastic Poisson process. Stochastic Process. Appl. 45 (1993) 331-349. | MR

[26] K. J. Utikal. Nonparametric inference for Markovian interval processes. Stochastic Process. Appl. 67 (1997) 1-23. | MR

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