ORACLE INEQUALITIES AND ADAPTIVE ESTIMATION IN THE CONVOLUTION STRUCTURE DENSITY MODEL

• 成果类型: Article
• 署名作者: Lepski, O., V; Willer, T.
• 署名单位: Aix-Marseille Universite
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/18-AOS1687
• 发表日期: 2019
• 页码: 233-287
• 关键词: Deconvolution; rates; CONVERGENCE; adaptation; selection; bounds

摘要

We study the problem of nonparametric estimation under L-p-loss, p is an element of[1, infinity), in the framework of the convolution structure density model on R-d. This observation scheme is a generalization of two classical statistical models, namely density estimation under direct and indirect observations. The original pointwise selection rule from a family of kernel-type estimators is proposed. For the selected estimator, we prove an L-p-norm oracle inequality and several of its consequences. Next, the problem of adaptive minimax estimation under L-p-loss over the scale of anisotropic Nikol'skii classes is addressed. We fully characterize the behavior of the minimax risk for different relationships between regularity parameters and norm indexes in the definitions of the functional class and of the risk. We prove that the proposed selection rule leads to the construction of an optimally or nearly optimally (up to logarithmic factors) adaptive estimator.