Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes

Romain Azaïs, François Dufour, and Anne Gégout-Petit

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Abstract

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.

Résumé

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.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 49, Number 4 (2013), 1204-1231.

Dates
First available in Project Euclid: 2 October 2013

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1380718744

Digital Object Identifier
doi:10.1214/12-AIHP503

Mathematical Reviews number (MathSciNet)
MR3127920

Zentralblatt MATH identifier
1293.62066

Subjects
Primary: 62G05: Estimation
Secondary: 62M09: Non-Markovian processes: estimation

Keywords
Non-homogeneous marked renewal process Nonparametric estimation Jump rate estimation Nelson–Aalen estimator Asymptotic consistency Ergodicity of Markov chains

Citation

Azaïs, Romain; Dufour, François; Gégout-Petit, Anne. Nonparametric estimation of the jump rate for non-homogeneous marked renewal processes. Ann. Inst. H. Poincaré Probab. Statist. 49 (2013), no. 4, 1204--1231. doi:10.1214/12-AIHP503. https://projecteuclid.org/euclid.aihp/1380718744


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