Adaptive minimax density estimation on ℝd for Huber’s contamination model

Abstract

Abstract We address the problem of adaptive minimax density estimation on $\mathbb{R}^{d}$ with $L_{p}$ loss functions under Huber’s contamination model. To investigate the contamination effect on the optimal estimation of the density, we first establish the minimax rate with the assumption that the density is in an anisotropic Nikol’skii class. We then develop a data-driven bandwidth selection procedure for kernel estimators, which can be viewed as a robust generalization of the Goldenshluger-Lepski method. We show that the proposed bandwidth selection rule can lead to the estimator being minimax adaptive to either the smoothness parameter or the contamination proportion. When both of them are unknown, we prove that finding any minimax-rate adaptive method is impossible. Extensions to smooth contamination cases are also discussed.