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We study the non-parametric estimation of an unknown density f with supporton R+ based on an i.i.d. sample with multiplicative measurement errors. Theproposed fully data driven procedure is based on the estimation of the Mellintransform of the density f , a regularisation of the inverse of the Mellintransform by a spectral cut-off and a data-driven model selection in order todeal with the upcoming bias-variance trade-off. We introduce and discussfurther Mellin-Sobolev spaces which characterize the regularity of the unknowndensity f through the decay of its Mellin transform. Additionally, we showminimax-optimality over Mellin-Sobolev spaces of the data-driven densityestimator and hence its adaptivity.

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