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June 2015 Adaptive estimation over anisotropic functional classes via oracle approach
Oleg Lepski
Ann. Statist. 43(3): 1178-1242 (June 2015). DOI: 10.1214/14-AOS1306


We address the problem of adaptive minimax estimation in white Gaussian noise models under $\mathbb{L}_{p}$-loss, $1\leq p\leq\infty$, on the anisotropic Nikol’skii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For the proposed estimator we establish so-called $\mathbb{L}_{p}$-norm oracle inequality and use it for deriving minimax adaptive results. We prove the existence of rate-adaptive estimators and fully characterize behavior of the minimax risk for different relationships between regularity parameters and norm indexes in definitions of the functional class and of the risk. In particular some new asymptotics of the minimax risk are discovered, including necessary and sufficient conditions for the existence of a uniformly consistent estimator. We provide also a detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation.


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Oleg Lepski. "Adaptive estimation over anisotropic functional classes via oracle approach." Ann. Statist. 43 (3) 1178 - 1242, June 2015.


Received: 1 April 2014; Revised: 1 November 2014; Published: June 2015
First available in Project Euclid: 15 May 2015

zbMATH: 1328.62213
MathSciNet: MR3346701
Digital Object Identifier: 10.1214/14-AOS1306

Primary: 62G05 , 62G20

Keywords: $\mathbb{L}_{p}$-risk , adaptive estimation , kernel estimators with varying bandwidths , Oracle inequality , White Gaussian noise model

Rights: Copyright © 2015 Institute of Mathematical Statistics


Vol.43 • No. 3 • June 2015
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