Adaptivity in convolution models with partially known noise distribution

Details Adaptivity in convolution models with partially known noise distribution Adaptivity in convolution models with partially known noise distribution

2008-04-07

arxiv.org/abs/0804.1056v2

Electronic Journal of Statistics 2008, Vol. 2, 897-915

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/08-EJS225 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by t

Scientific paper

10.1214/08-EJS225

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 $\psi_n^{-2}\|f-f_0\|_2^2\ge \mathcal{C}$). 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.

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