An adaptive procedure for Fourier estimators: illustration 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.
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Pré-publication, Document de travail
MAP5 2018-04. 2018
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https://hal.archives-ouvertes.fr/hal-01700525
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Dernière modification le : mercredi 21 février 2018 - 01:10:58
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  • HAL Id : hal-01700525, version 1
  • ARXIV : 1802.05104

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Céline Duval, Johanna Kappus. An adaptive procedure for Fourier estimators: illustration to deconvolution and decompounding. MAP5 2018-04. 2018. 〈hal-01700525〉

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