## The Annals of Statistics

### Oracle inequalities and adaptive estimation in the convolution structure density model

#### Abstract

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.

#### Article information

Source
Ann. Statist., Volume 47, Number 1 (2019), 233-287.

Dates
Revised: November 2017
First available in Project Euclid: 30 November 2018

https://projecteuclid.org/euclid.aos/1543568588

Digital Object Identifier
doi:10.1214/18-AOS1687

Mathematical Reviews number (MathSciNet)
MR3909933

Zentralblatt MATH identifier
07036201

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties

#### Citation

Lepski, O. V.; Willer, T. Oracle inequalities and adaptive estimation in the convolution structure density model. Ann. Statist. 47 (2019), no. 1, 233--287. doi:10.1214/18-AOS1687. https://projecteuclid.org/euclid.aos/1543568588

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