We study the non-parametric estimation of an unknown density f with support on based on an i.i.d. sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the density f, a regularisation of the inverse of the Mellin transform by a spectral cut-off and a data-driven model selection in order to deal with the upcoming bias-variance trade-off. We introduce and discuss further Mellin-Sobolev spaces which characterize 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 data-driven density estimator and hence its adaptivity.
"Spectral cut-off regularisation for density estimation under multiplicative measurement errors." Electron. J. Statist. 15 (1) 3551 - 3573, 2021. https://doi.org/10.1214/21-EJS1870