Adaptive Density Estimation on Bounded Domains

TL;DR

This paper introduces a new adaptive kernel density estimation method for bounded domains that effectively addresses boundary bias and adapts to various smoothness classes without prior parameter knowledge.

Contribution

It proposes a novel family of boundary bias-free kernel estimators and a data-driven selection procedure that adaptively chooses kernels and bandwidths based on the Goldenshluger-Lepski method.

Findings

Establishes oracle inequalities for the proposed estimators.

Demonstrates adaptivity over anisotropic and isotropic Sobolev-Slobodetskii classes.

Achieves boundary bias correction without smoothness restrictions.

Abstract

We study the estimation, in Lp-norm, of density functions defined on [0,1]^d. We construct a new family of kernel density estimators that do not suffer from the so-called boundary bias problem and we propose a data-driven procedure based on the Goldenshluger and Lepski approach that jointly selects a kernel and a bandwidth. We derive two estimators that satisfy oracle-type inequalities. They are also proved to be adaptive over a scale of anisotropic or isotropic Sobolev-Slobodetskii classes (which are particular cases of Besov or Sobolev classical classes). The main interest of the isotropic procedure is to obtain adaptive results without any restriction on the smoothness parameter.