Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

Abstract

The purpose of this paper is twofold. First, introduce a new adaptive procedure to select the optimal – up to a logarithmic factor – cutoff parameter for Fourier density estimators. Two inverse problems are considered: deconvolution and decompounding. Deconvolution is a typical inverse problem, for which our procedure is numerically simple and stable, a comparison is performed with penalized techniques. Moreover, the procedure and the proof of oracle bounds do not rely on any knowledge on the noise term. Second, for decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of $n$ increments at timestep $\Delta$, build an unified adaptive estimator which is optimal – up to a logarithmic factor – regardless the behavior of $\Delta$.