Adaptive estimation over anisotropic functional classes via oracle approach

TL;DR

This paper develops a data-driven kernel estimation method for adaptive minimax estimation over anisotropic Nikolskii classes in Gaussian noise, establishing oracle inequalities and characterizing minimax risk behavior.

Contribution

It introduces a new adaptive estimation procedure with a selection scheme, deriving minimax results and asymptotics for anisotropic functional classes.

Findings

Existence of rate-adaptive estimators under various conditions

New asymptotics of the minimax risk identified

Conditions for uniformly consistent estimators established

Abstract

We address the problem of adaptive minimax estimation in white gaussian noise model under $L_p$--loss, $1\leq p\leq\infty,$ on the anisotropic Nikolskii classes. We present the estimation procedure based on a new data-driven selection scheme from the family of kernel estimators with varying bandwidths. For proposed estimator we establish so-called Lp-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 existence a uniformly consistent estimator. We provide also with detailed overview of existing methods and results and formulate open problems in adaptive minimax estimation.