Anisotropic spectral cut-off estimation under multiplicative measurement errors

Abstract

We study the non-parametric estimation of an unknown density f with support on R+d 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 and a regularization of the inverse of Mellin transform by a spectral cut-off. The bias–variance tradeoff for estimating f is optimized with a data-driven anisotropic choice of the cutoff parameter. In order to discuss the bias term, we consider the Mellin–Sobolev spaces which define 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 spectral cut-off density estimator.