An adaptive procedure for Fourier estimators: illustration to deconvolution and decompounding
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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 Δ > 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 Δ that can vanish, Δ := Δ n → 0, can be fixed, Δ n → Δ 0 > 0 or can get large, Δ 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.
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