Adaptive procedure for Fourier estimators: application to deconvolution and decompounding

Abstract

We introduce a new procedure to select the optimal cutoff parameter for Fourier density estimators that leads to adaptive rate optimal estimators, up to a logarithmic factor. This adaptive procedure applies for different inverse problems. We illustrate it on two classical examples: deconvolution and decompounding, i.e. non-parametric estimation of the jump density of a compound Poisson process from the observation of n increments of length ∆ > 0. For this latter example, we first build an estimator for which we provide an upper bound for its L 2 -risk that is valid simultaneously for sampling rates ∆ that can vanish, ∆ := ∆ n → 0, can be fixed, ∆ n → ∆ 0 > 0 or can get large, ∆ n → ∞ slowly. This last result is new and presents interest on its own. Then, we show that the adaptive procedure we present leads to an adaptive and rate optimal (up to a logarithmic factor) estimator of the jump density.

Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

“…Duval (2014) studies statistical inference on compound Poisson processes under macroscopic observations. Duval and Kappus (2018) is another recent study on nonparametric estimation on compound Poisson processes under macroscopic observations. Coca (2018b) discusses the robustness of spectral estimation of Lévy measures of compound Poisson processes to ∆ n , and it includes the consistency of the estimator under the macroscopic set up.…”