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 $\Delta$.
Highlights
In the literature on non-parametric statistics a lot of space is dedicated to adaptive procedures
Adaptivity may be understood as minimax-adaptivity, i.e. optimal rates of convergence are attained simultaneously over a collection of class of densities, such as Sobolev-balls
We propose an approach that is relevant for inverse problems when the estimator relies on Fourier techniques
Summary
In the literature on non-parametric statistics a lot of space is dedicated to adaptive procedures. Adaptivity may refer to proving nonasymptotic oracle bounds, i.e. having a procedure that mimics, up to a constant, the estimator that minimizes a given loss function. It is this last notion of adaptivity we adopt here. An advantage of the procedure presented here is that, in the cases considered, for all the values of κ such that the oracle bound bound is valid, the corresponding estimator is reasonable Many adaptive procedures such as penalization methods minimizes an empirical version of the upper bound (1.2), while the spirit of (1.4) consists in finding the zeroes of an empirical version of the derivative in m of the upper bound (1.2). We compare our results with a penalization procedure described in Comte and Lacour [12], which are known to be rapid and efficient in deconvolution contexts
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