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.

翻译：我们研究了在未知密度函数下评估的线性功能值的非参数估算值 $ {theta} (f) 。 依据一. d. 样本, 以美元为单位, 支持 $ $ $ $ 美元 。 提议的估算程序结合了对Mellin 密度变换的估算和通过光谱截断对Mellin 变换反向的常规化。 为了控制平均正方差, 我们区分了通过即将到来的Mellin 变换和线性功能平滑的不同衰减而发现的几种情景。 事实上, 我们确定了一些情景, 在哪些情景中, 需要非三边选择即将到来的调试参数, 并基于Goldenshluger- Lepski 方法提出数据驱动选择。 此外, 我们展示了测量器Mellin- Sobolev 空间的微质量- 最佳性。