文章基本信息

标题：Density deconvolution with non–standard error distributions: Rates of convergence and adaptive estimation
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作者：Alexander Goldenshluger ; Taeho Kim
期刊名称：Electronic Journal of Statistics
印刷版ISSN：1935-7524
出版年度：2021
卷号：15
期号：1
页码：3394-3427
DOI：10.1214/21-EJS1863
语种：English
出版社：Institute of Mathematical Statistics
摘要：It is a 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.
关键词：62G07; 62G20; Characteristic function; density deconvolution; Laplace transform; minimax risk; non-standard measurement error; zero multiplicity