Minimax properties of beta kernel density estimators

TL;DR

This paper investigates the asymptotic minimax properties of beta kernel density estimators, highlighting their boundary effect advantages and limitations for certain regularities and loss functions.

Contribution

It provides a theoretical analysis showing when beta kernel estimators are minimax and when they are not, depending on function regularity and loss types.

Findings

Beta kernel estimators are minimax for regularities of order two or less.

They are not minimax for very regular functions or certain loss functions.

Beta kernels effectively handle boundary effects in density estimation.

Abstract

In this paper, we are interested in the study of beta kernel estimators from an asymptotic minimax point of view. It is well known that beta kernel estimators are, on the contrary of classical kernel estimators, "free of boundary effect" and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.