Density deconvolution with non–standard error distributions: Rates of convergence and adaptive estimation
Density deconvolution with non–standard error distributions: Rates of convergence and adaptive estimation
复制标题
具有非标准误差分布的密度反卷积：收敛率和自适应估计
DOI：
10.1214/21-ejs1863
复制
发表时间：
2021
影响因子：
1.1
通讯作者：
Taeho Kim
中科院分区：
文献类型：
--
作者：
A. Goldenshluger;Taeho Kim
在密度反褶积问题中，测量误差分布的特征函数在实线上为非零是一个典型的标准假设。虽然大多数现有的关于该主题的著作都假定了这一条件，但也有许多令人感兴趣的问题实例违反了这一条件。本文针对测量误差特征函数为零的非标准设置，研究了零点重数对估计精度的影响。对于这类典型问题，我们证明了最佳估计精度是由零点的重数、误差特征函数的衰减率以及估计密度的光滑性和尾部性质决定的。我们得到了极小极大风险的下界，并在极小极大意义下得到了最优估计。此外，我们还考虑了自适应估计问题，提出了一种数据驱动估计器，该估计器能够自动适应待估计密度的未知光滑性和尾部行为。
It is a typical standard assumption in the density deconvolution problem that the characteristic function of the measurement error distribution is non-zero on the real line. While this condition is assumed in the majority of existing works on the topic, there are many problem instances of interest where it is violated. In this paper we focus on non--standard settings where the characteristic function of the measurement errors has zeros, and study how zeros multiplicity affects the estimation accuracy. For a prototypical problem of this type we demonstrate that the best achievable estimation accuracy is determined by the multiplicity of zeros, the rate of decay of the error characteristic function, as well as by the smoothness and the tail behavior of the estimated density. We derive lower bounds on the minimax risk and develop optimal in the minimax sense estimators. In addition, we consider the problem of adaptive estimation and propose a data-driven estimator that automatically adapts to unknown smoothness and tail behavior of the density to be estimated.