Section: New Results
Estimation of the jump rate of a PDMP
Participants : Romain Azaïs, François Dufour, Anne Gégout-Petit.
We estimate the jump rate of PDMP. We suppose the flow given by physics laws and we want to make some inference on . being deterministic, the problem can be rewritten as a problem of estimation of the rate with with an open set of a separable metric space. We have an ergodicity assumption on the observed PDMP and the asymptotic is in the time of observation of the process.
We distinguish three cases :
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is finite. In this case, we easily estimate each of the cumulated risk functions corresponding to each of by a Nelson Aalen estimator. The results is based on the decomposition in semi-martingale of the following counting process in an appropriate filtration:
We obtain the estimator of the rate by smoothing of the estimator of .
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is an open set of a general separable metric space but the transition measure does not depend on the time spent in the current regime. In this case, we suppose the rate Lipschitz and the process ergodic with a stationary law denoted by . We first construct an estimation of the cumulated rate knowing that belongs to a set such that by :
We show the consistence of the estimator. Smoothing and using a fine partition of allow us to obtain an uniform result for the approximation of the rate , in some sense in and .
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is an open set of a general separable metric space and the transition measure depends on the time spent in the current regime. Here, we loose some conditional independence between the 's and the whole set of the locations of the jump . We have to make a detour for the estimation of the law of the time knowing the current by the the law knowing . The method gives an estimation of the conditional density of given .
We have made simulation studies that give expected results. A R package for this estimation method is in progress.
This work is a part of the PhD Thesis of R. Azaïs founded by the ANR Fautocoes. R. Azaïs has presented a part of this work at "Rencontres des Jeunes Statisticiens" in 2011 September [28] . The work will be soon submitted to a international peer-reviewed journal for publication.