Estimation of the Jump Size Density in a Mixed Compound Poisson Process

Abstract. Consider a mixed compound process Y (t) = ∑N(Λt) i=1 ξi where N is a Poisson process with intensity 1, Λ a positive random variable, (ξi) a sequence of i.i.d. random variables with density f and (N,Λ, (ξi)) are independent. In this paper, we study nonparametric estimators of f by specific deconvolution methods. Assuming that Λ has exponential distribution with unknown expectation, we propose two types of estimators based on the observation of an i.i.d. sample (Yj(∆))1≤j≤n for ∆ a given time. One strategy is for fixed ∆, the other for small ∆ (with large n∆). Risks bounds and adaptive procedures are provided. Then, with no assumption on the distribution of Λ, we propose a nonparametric estimator of f based on the joint observation (Nj(Λj∆), Yj(∆))1≤j≤n. Risks bounds are provided leading to unusual rates. The methods are implemented and compared via simulations. October 2, 2014