标题： 在存在依赖性的情况下自适应非参数估计 显示英文标题

标题： Adaptive non-parametric estimation in the presence of dependence

摘要： 我们研究了存在相依数据时的非参数估计问题，特别是随机设计下的非参数回归和非参数密度估计。 所提出的估计程序基于维度约简。 假设混合系数快速递减以表征足够弱的依赖性，推导出了该估计量的渐近最优收敛速度。 我们通过经典的光滑性假设来说明这些结果。 然而，所提出的估计器需要根据感兴趣的函数的某些特性（这些特性在实际中未知）选择一个最佳的维度参数。 我们在工作中解决的主要问题是结合模型选择和Lepski方法的自适应选择这一维度参数。 它受到Goldenshluger和Lepski（2011年）最近工作的启发。 我们证明，只要混合系数衰减迅速，这种数据驱动的估计器可以达到下界风险至常数倍。 显示英文摘要

摘要： We consider non-parametric estimation problems in the presence of dependent data, notably non-parametric regression with random design and non-parametric density estimation. The proposed estimation procedure is based on a dimension reduction. The minimax optimal rate of convergence of the estimator is derived assuming a sufficiently weak dependence characterized by fast decreasing mixing coefficients. We illustrate these results by considering classical smoothness assumptions. However, the proposed estimator requires an optimal choice of a dimension parameter depending on certain characteristics of the function of interest, which are not known in practice. The main issue addressed in our work is an adaptive choice of this dimension parameter combining model selection and Lepski's method. It is inspired by the recent work of Goldenshluger and Lepski (2011). We show that this data-driven estimator can attain the lower risk bound up to a constant provided a fast decay of the mixing coefficients.