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Last edited on
Oct 10, 2022 by JJ
.

Discussion paper:
arXiv:2111.14920

Title:
Linear functional estimation under multiplicative measurement errors

Authors:
Sergio Brenner Miguel, Fabienne Comte (Université Paris Descartes) and Jan JOHANNES

Abstract:
We study the non-parametric estimation of the value θ(f) of a linear functional evaluated at an unknown density function f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed estimation procedure combines the estimation of the Mellin transform of the density f and a regularisation of the inverse of the Mellin transform by a spectral cut-off. In order to bound the mean squared error we distinguish several scenarios characterised through different decays of the upcoming Mellin transforms and the smoothnes of the linear functional. In fact, we identify scenarios, where a non-trivial choice of the upcoming tuning parameter is necessary and propose a data-driven choice based on a Goldenshluger-Lepski method. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the estimator.

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