Section: New Results
Estimation of the jump rate of a PDMP
Participants : Romain Azaïs, François Dufour, Anne GégoutPetit.
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 $\lambda (z,t)$ 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 :

$E$ is finite. In this case, we easily estimate each of the cumulated risk functions $\Lambda (z,t)=exp({\int}_{0}^{t}\lambda (z,s)ds)$ corresponding to each of $z\in E$ by a Nelson Aalen estimator. The results is based on the decomposition in semimartingale of the following counting process in an appropriate filtration:
$\forall t\ge 0,\phantom{\rule{3.33333pt}{0ex}}{N}_{n}(z,t)=\sum _{i=0}^{n1}{\mathbf{1}}_{\{{S}_{i+1}\le t\}}{\mathbf{1}}_{\{{Z}_{i}=z\}},$ (1) We obtain the estimator of the rate $\lambda (z,t)$ by smoothing of the estimator of $\Lambda $.

$E$ 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 $\lambda (z,t)$ 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 :
${\widehat{L}}_{n}(A,t)=\sum _{i=0}^{n1}\frac{1}{{Y}_{n}(A,{S}_{i+1})}{\mathbf{1}}_{\{{S}_{i+1}\le t\}}{\mathbf{1}}_{\{{Z}_{i}\in A\}}\phantom{\rule{2.em}{0ex}}\text{with}\phantom{\rule{2.em}{0ex}}{Y}_{n}(A,t)=\sum _{i=0}^{n1}{\mathbf{1}}_{\{{S}_{i+1}\ge t\}}{\mathbf{1}}_{\{{Z}_{i}\in A\}}.$ (2) We show the consistence of the estimator. Smoothing ${\widehat{L}}_{n}(A,t)$ and using a fine partition of $E$ allow us to obtain an uniform result for the approximation of the rate $\lambda (z,t)$, in some sense in $t$ and $z$.

$E$ 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 $\{{Z}_{1},...,{Z}_{n}\}$. 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 $({Z}_{k},{Z}_{k+1})$. 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 peerreviewed journal for publication.