  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.
