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

Fabienne Comte (1) , Céline Duval , Valentine Genon-Catalot (1) , Johanna Kappus

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

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.

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

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