We study the non-parametric estimation of an unknown density f with support on R+ based on an i.i.d. sample with multiplicative measurement errors. The proposed fully-data driven procedure consists of the estimation of the Mellin transform of the density f and a regularisation of the inverse of the Mellin transform by a ridge approach. The upcoming bias-variance trade-off is dealt with by a data-driven choice of the ridge parameter. In order to discuss the bias term, we consider the Mellin-Sobolev spaces which characterise 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 ridge density estimator.

翻译：我们研究对未知密度的非参数估计,在 R+ 上提供支持, 其依据是 i.d. 样本, 并带有多复制度测量错误。 拟议的全数据驱动程序包括估算Mellin 密度 f 的变异和以脊法对Mellin 变异的反向进行常规化。 即将到来的偏差权衡由数据驱动的脊参数选择处理。 为了讨论偏差术语, 我们考虑Mellin- Sobolev 空间, 其特征是未知密度f 的规律性, 贯穿其Mellin 变异的衰变。 此外, 我们展示了脊密度天体的Mellin- Sobolev 空间的微质量- 最佳性 。