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## 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 $\lambda$. $\phi$ being deterministic, the problem can be rewritten as a problem of estimation of the rate $\lambda \left(z,t\right)$ with $z\in E$ with $E$ 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 :

1. $E$ is finite. In this case, we easily estimate each of the cumulated risk functions $\Lambda \left(z,t\right)=exp\left(-{\int }_{0}^{t}\lambda \left(z,s\right)ds\right)$ corresponding to each of $z\in E$ by a Nelson Aalen estimator. The results is based on the decomposition in semi-martingale of the following counting process in an appropriate filtration:

 $\forall t\ge 0,\phantom{\rule{3.33333pt}{0ex}}{N}_{n}\left(z,t\right)=\sum _{i=0}^{n-1}{\mathbf{1}}_{\left\{{S}_{i+1}\le t\right\}}{\mathbf{1}}_{\left\{{Z}_{i}=z\right\}},$ (1)

We obtain the estimator of the rate $\lambda \left(z,t\right)$ by smoothing of the estimator of $\Lambda$.

2. $E$ is an open set of a general separable metric space but the transition measure $Q$ does not depend on the time spent in the current regime. In this case, we suppose the rate $\lambda \left(z,t\right)$ Lipschitz and the process ergodic with a stationary law denoted by $\nu$. We first construct an estimation of the cumulated rate knowing that $z$ belongs to a set $A$ such that $\nu \left(A\right)>0$ by :

 ${\stackrel{^}{L}}_{n}\left(A,t\right)=\sum _{i=0}^{n-1}\frac{1}{{Y}_{n}\left(A,{S}_{i+1}\right)}{\mathbf{1}}_{\left\{{S}_{i+1}\le t\right\}}{\mathbf{1}}_{\left\{{Z}_{i}\in A\right\}}\phantom{\rule{2.em}{0ex}}\text{with}\phantom{\rule{2.em}{0ex}}{Y}_{n}\left(A,t\right)=\sum _{i=0}^{n-1}{\mathbf{1}}_{\left\{{S}_{i+1}\ge t\right\}}{\mathbf{1}}_{\left\{{Z}_{i}\in A\right\}}.$ (2)

We show the consistence of the estimator. Smoothing ${\stackrel{^}{L}}_{n}\left(A,t\right)$ and using a fine partition of $E$ allow us to obtain an uniform result for the approximation of the rate $\lambda \left(z,t\right)$, in some sense in $t$ and $z$.

3. $E$ is an open set of a general separable metric space and the transition measure $Q$ depends on the time spent in the current regime. Here, we loose some conditional independence between the ${S}_{i}$'s and the whole set of the locations of the jump $\left\{{Z}_{1},...,{Z}_{n}\right\}$. We have to make a detour for the estimation of the law of the time ${S}_{k+1}$ knowing the current ${Z}_{k}$ by the the law ${S}_{k+1}$ knowing $\left({Z}_{k},{Z}_{k+1}\right)$. The method gives an estimation of the conditional density of ${S}_{k+1}$ given ${Z}_{k}$.

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.