  We study the non-parametric estimation of an unknown density f with support
on R+ based on an i.i.d. sample with multiplicative measurement errors. The
proposed fully-data driven procedure consists of the estimation of the Mellin
transform of the density f and a regularisation of the inverse of the Mellin
transform by a ridge approach. The upcoming bias-variance trade-off is dealt
with by a data-driven choice of the ridge parameter. In order to discuss the
bias term, we consider the Mellin-Sobolev spaces which characterise the
regularity of the unknown density f through the decay of its Mellin transform.
Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the
ridge density estimator.
