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 $\lambda (z,t)$ 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 :

1. $E$ 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 semi-martingale of the following counting process in an appropriate filtration:

   $\forall t\ge 0,N_n(z,t)=\sum _{i=0}^{n-1}{\mathbf{1}}_{\{S_{i+1}\le t\}}{\mathbf{1}}_{\{Z_i=z\}},$

   (1)

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

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 $\lambda (z,t)$ 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 :

   ${\widehat{L}}_n(A,t)=\sum _{i=0}^{n-1}\frac{1}{Y_n(A,S_{i+1})}{\mathbf{1}}_{\{S_{i+1}\le t\}}{\mathbf{1}}_{\{Z_i\in A\}}\text{with}{Y}_n(A,t)=\sum _{i=0}^{n-1}{\mathbf{1}}_{\{S_{i+1}\ge t\}}{\mathbf{1}}_{\{Z_i\in A\}}.$

   (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 $z$.

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 $\{Z_1,...,Z_n\}$. 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 peer-reviewed journal for publication.