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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     title = {Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes},
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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/

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