Adaptive estimation over anisotropic functional classes via oracle approach

Abstract

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.