We consider a multiplicative deconvolution problem, in which the density $f$ or the survival function $S^X$ of a strictly positive random variable $X$ is estimated nonparametrically based on an i.i.d. sample from a noisy observation $Y = X\cdot U$ of $X$. The multiplicative measurement error $U$ is supposed to be independent of $X$. The objective of this work is to construct a fully data-driven estimation procedure when the error density $f^U$ is unknown. We assume that in addition to the i.i.d. sample from $Y$, we have at our disposal an additional i.i.d. sample drawn independently from the error distribution. The proposed estimation procedure combines the estimation of the Mellin transformation of the density $f$ and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The derived risk bounds and oracle-type inequalities cover both - the estimation of the density $f$ as well as the survival function $S^X$. The main issue addressed in this work is the data-driven choice of the cut-off parameter using a model selection approach. We discuss conditions under which the fully data-driven estimator can attain the oracle-risk up to a constant without any previous knowledge of the error distribution. We compute convergences rates under classical smoothness assumptions. We illustrate the estimation strategy by a simulation study with different choices of distributions.

翻译：我们考虑一个乘法反卷积问题，其中基于来自含噪观测 $Y = X\cdot U$ 的独立同分布样本，对严格正随机变量 $X$ 的密度 $f$ 或生存函数 $S^X$ 进行非参数估计。乘法测量误差 $U$ 被假定与 $X$ 独立。本文的目标是在误差密度 $f^U$ 未知的情况下，构建一种完全数据驱动的估计方法。我们假设，除了来自 $Y$ 的独立同分布样本外，我们还拥有一个独立于误差分布的额外独立同分布样本。所提出的估计方法结合了密度 $f$ 的梅林变换估计以及通过谱截断对梅林逆变换的正则化。导出的风险界和预言不等式同时涵盖了密度 $f$ 和生存函数 $S^X$ 的估计。本文解决的主要问题是通过模型选择方法对截断参数进行数据驱动选择。我们讨论了在全数据驱动估计器无需误差分布任何先验知识即可达到常数倍预言风险的条件。我们在经典光滑性假设下计算收敛速度。通过不同分布选择的模拟研究来说明估计策略。