We study the non-parametric estimation of an unknown survival function S with support on R+ based on a sample with multiplicative measurement errors. The proposed fully-data driven procedure is based on the estimation of the Mellin transform of the survival function S and a regularisation of the inverse of the Mellin transform by a spectral cut-off. The upcoming bias-variance trade-off is dealt with by a data-driven choice of the cut-off parameter. In order to discuss the bias term, we consider the Mellin-Sobolev spaces which characterize the regularity of the unknown survival function S through the decay of its Mellin transform. For the analysis of the variance term, we consider the i.i.d. case and incorporate dependent observations in form of Bernoulli shift processes and beta mixing sequences. Additionally, we show minimax-optimality over Mellin-Sobolev spaces of the spectral cut-off estimator.
翻译:我们研究一个未知生存函数S的非参数估计,在R+的支持下,基于一个具有多复制度测量误差的样本。提议的完全数据驱动程序基于对生存函数S的Mellin变换的估计,以及用光谱截断法对Mellin变换的反面进行常规化。即将到来的偏差权衡是通过数据驱动选择断线参数来处理的。为了讨论偏差术语,我们考虑了Mellin-Sobolev空间,该空间通过Mellin变换法的衰减来说明未知生存函数S的规律性。关于变换术语的分析,我们考虑i.i.d.案例,并以Bernoulli变换过程和乙型混合序列的形式纳入依赖性观测。此外,我们展示了光谱切断点测量器的Mellin-Sobolev空间的微质量。