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February 2019 Oracle inequalities and adaptive estimation in the convolution structure density model
O. V. Lepski, T. Willer
Ann. Statist. 47(1): 233-287 (February 2019). DOI: 10.1214/18-AOS1687


We study the problem of nonparametric estimation under $\mathbb{L}_{p}$-loss, $p\in[1,\infty)$, in the framework of the convolution structure density model on $\mathbb{R}^{d}$. 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 $\mathbb{L}_{p}$-norm oracle inequality and several of its consequences. Next, the problem of adaptive minimax estimation under $\mathbb{L}_{p}$-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.


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O. V. Lepski. T. Willer. "Oracle inequalities and adaptive estimation in the convolution structure density model." Ann. Statist. 47 (1) 233 - 287, February 2019.


Received: 1 April 2017; Revised: 1 November 2017; Published: February 2019
First available in Project Euclid: 30 November 2018

zbMATH: 07036201
MathSciNet: MR3909933
Digital Object Identifier: 10.1214/18-AOS1687

Primary: 62G05 , 62G20

Keywords: $\mathbb{L}_{p}$-risk , adaptive estimation , anisotropic Nikol’skii class , Deconvolution model , Density estimation , kernel estimators , Oracle inequality

Rights: Copyright © 2019 Institute of Mathematical Statistics


Vol.47 • No. 1 • February 2019
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