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A recursive nonparametric estimator for the transition kernel of a piecewise-deterministic Markov process

Romain Azaïs — 2014

ESAIM: Probability and Statistics

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...

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

Romain Azaïs;  François Dufour;  Anne Gégout-Petit — 2013

Annales de l'I.H.P. Probabilités et statistiques

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...

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