On adaptive minimax density estimation on $R^d$

TL;DR

This paper investigates the fundamental limits of adaptive minimax density estimation on multi-dimensional real space, providing a comprehensive characterization of the minimax risk across different regularity regimes and proposing a nearly optimal adaptive estimator.

Contribution

It offers a full characterization of the minimax risk behavior for anisotropic Nikol'skii classes and introduces a data-driven kernel estimator that is nearly optimal across all regimes.

Findings

Identifies four distinct regimes of minimax risk behavior.

Provides a single estimator that adapts nearly optimally across all regimes.

Characterizes the influence of regularity parameters and norm indexes on estimation risk.

Abstract

We address the problem of adaptive minimax density estimation on $\bR^d$ with $\bL_p$--loss on the anisotropic Nikol'skii classes. We 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, we show that there are four different regimes with respect to the behavior of the minimax risk. We develop a single estimator which is (nearly) optimal in orderover the complete scale of the anisotropic Nikol'skii classes. Our estimation procedure is based on a data-driven selection of an estimator from a fixed family of kernel estimators.