# Estimation of the jump size density in a mixed compound Poisson process

Abstract : Consider a mixed compound process $Y(t)=\sum_{i=1}^{N(\Lambda t)}\xi_i$ where $N$ is a Poisson process with intensity 1, $\Lambda$ a positive random variable, $(\xi_i)$ a sequence of {\em i.i.d.} random variables with density $f$ and $(N,\Lambda,(\xi_i))$ are independent. In this paper, we study nonparametric estimators of $f$ by specific deconvolution methods. Assuming that $\Lambda$ has exponential distribution with unknown expectation, we propose two types of estimators based on the observation of an {\em i.i.d.} sample $(Y_j(\Delta))_{1\leq j\leq n}$ for $\Delta$ a given time. One strategy is for fixed $\Delta$, the other for small $\Delta$ (with large $n\Delta$). Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of $\Lambda$, we propose a nonparametric estimator of $f$ based on the joint observation $(N_j(\Lambda_j\Delta), Y_j(\Delta))_{1\leq j\leq n}$. Risks bounds are provided leading to unusual rates. The methods are implemented and compared via simulations.
Type de document :
Pré-publication, Document de travail
MAP5 2014-13. 2014
Domaine :

https://hal.archives-ouvertes.fr/hal-00995037
Contributeur : Fabienne Comte <>
Soumis le : mardi 14 octobre 2014 - 13:32:11
Dernière modification le : mardi 11 octobre 2016 - 13:29:55

### Annexes

MixedCompoundSubm.pdf
Fichiers produits par l'(les) auteur(s)

### Identifiants

• HAL Id : hal-00995037, version 2

### Citation

Fabienne Comte, Céline Duval, Valentine Genon-Catalot, Johanna Kappus. Estimation of the jump size density in a mixed compound Poisson process. MAP5 2014-13. 2014. <hal-00995037v2>