Linear functional estimation under multiplicative measurement errors

TL;DR

This paper develops a new non-parametric estimator for linear functionals of unknown densities with multiplicative measurement errors, using Mellin transforms and spectral regularisation, achieving minimax optimality.

Contribution

It introduces a novel estimation method combining Mellin transform estimation with spectral cut-off and data-driven tuning, improving accuracy under multiplicative errors.

Findings

Estimator achieves minimax optimality in Mellin-Sobolev spaces.

Proposes a data-driven tuning method based on Goldenshluger-Lepski.

Performs well across different decay and smoothness scenarios.

Abstract

We study the non-parametric estimation of the value ${\theta}(f )$ 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.