  We consider a multiplicative deconvolution problem, in which the density $f$
or the survival function $S^X$ of a strictly positive random variable $X$ is
estimated nonparametrically based on an i.i.d. sample from a noisy observation
$Y = X\cdot U$ of $X$. The multiplicative measurement error $U$ is supposed to
be independent of $X$. The objective of this work is to construct a fully
data-driven estimation procedure when the error density $f^U$ is unknown. We
assume that in addition to the i.i.d. sample from $Y$, we have at our disposal
an additional i.i.d. sample drawn independently from the error distribution.
The proposed estimation procedure combines the estimation of the Mellin
transformation of the density $f$ and a regularisation of the inverse of the
Mellin transform by a spectral cut-off. The derived risk bounds and oracle-type
inequalities cover both - the estimation of the density $f$ as well as the
survival function $S^X$. The main issue addressed in this work is the
data-driven choice of the cut-off parameter using a model selection approach.
We discuss conditions under which the fully data-driven estimator can attain
the oracle-risk up to a constant without any previous knowledge of the error
distribution. We compute convergences rates under classical smoothness
assumptions. We illustrate the estimation strategy by a simulation study with
different choices of distributions.
