Spectral cut-off regularisation for density estimation under multiplicative measurement errors

We study the non-parametric estimation of an unknown density f with support on R+ based i.i.d. sample multiplicative measurement errors. The proposed fully-data driven procedure is Mellin transform f, a regularisation inverse by spectral cut-off and data-driven model selection in order to deal upcoming bias-variance trade-off. introduce discuss further Mellin-Sobolev spaces which characterize regularity through decay its transform. Additionally, we show minimax-optimality over estimator hence adaptivity.