# Adaptivity in convolution models with partially known noise distribution - Mathematics > Statistics Theory

Abstract: We consider a semiparametric convolution model. We observe random variableshaving a distribution given by the convolution of some unknown density $f$ andsome partially known noise density $g$. In this work, $g$ is assumedexponentially smooth with stable law having unknown self-similarity index $s$.In order to ensure identifiability of the model, we restrict our attention topolynomially smooth, Sobolev-type densities $f$, with smoothness parameter$\beta$. In this context, we first provide a consistent estimation procedurefor $s$. This estimator is then plugged-into three different procedures:estimation of the unknown density $f$, of the functional $\int f^2$ andgoodness-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$\psi n^{-2}\|f-f 0\| 2^2\ge \mathcal{C}$. These procedures are adaptive withrespect to both $s$ and $\beta$ and attain the rates which are known optimalfor known values of $s$ and $\beta$. As a by-product, when the noise density isknown and exponentially smooth our testing procedure is optimal adaptive fortesting Sobolev-type densities. The estimating procedure of $s$ is illustratedon synthetic data.

Author: Cristina Butucea, Catherine Matias, Christophe Pouet

Source: https://arxiv.org/