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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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