  We study the non-parametric estimation of an unknown survival function S with
support on R+ based on a sample with multiplicative measurement errors. The
proposed fully-data driven procedure is based on the estimation of the Mellin
transform of the survival function S and a regularisation of the inverse of the
Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is
dealt with by a data-driven choice of the cut-off parameter. In order to
discuss the bias term, we consider the Mellin-Sobolev spaces which characterize
the regularity of the unknown survival function S through the decay of its
Mellin transform. For the analysis of the variance term, we consider the i.i.d.
case and incorporate dependent observations in form of Bernoulli shift
processes and beta mixing sequences. Additionally, we show minimax-optimality
over Mellin-Sobolev spaces of the spectral cut-off estimator.
