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Adaptivity in convolution models with partially known noise distribution. (English) Zbl 1320.62066
Summary: We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $$f$$ and some partially known noise density $$g$$. In this work, $$g$$ is assumed exponentially smooth with stable law having unknown self-similarity index $$s$$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $$f$$, with smoothness parameter $$\beta$$. In this context, we first provide a consistent estimation procedure for $$s$$. This estimator is then plugged-into three different procedures: estimation of the unknown density $$f$$, of the functional $$\int f^{2}$$ and goodness-of-fit test of the hypothesis $$H_{0}:f=f_{0}$$, where the alternative $$H_{1}$$ is expressed with respect to $$\mathbb L_2$$-norm (i.e. has the form $$\psi_n^{-2}\|f-f_0\|^2_2\geq \mathcal C$$). These procedures are adaptive with respect to both $$s$$ and $$\beta$$ and attain the rates which are known optimal for known values of $$s$$ and $$\beta$$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of $$s$$ is illustrated on synthetic data.

##### MSC:
 62G05 Nonparametric estimation 62F12 Asymptotic properties of parametric estimators 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference
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##### References:
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