A new adaptive local polynomial density estimation procedure on complicated domains

TL;DR

This paper introduces a flexible, adaptive local polynomial density estimation method for complex domains, demonstrating minimax optimality and superior performance over existing methods through simulations and implementation in an R package.

Contribution

It develops a novel adaptive local polynomial density estimator applicable to complicated domains with theoretical guarantees and practical implementation.

Findings

Estimator is minimax under $L^2$ risk across H"older classes

Oracle inequalities and explicit convergence rates derived

Outperforms existing methods in simulations on polynomial sectors

Abstract

This paper presents a novel approach for pointwise estimation of multivariate density functions on known domains of arbitrary dimensions using nonparametric local polynomial estimators. Our method is highly flexible, as it applies to both simple domains, such as open connected sets, and more complicated domains that are not star-shaped around the point of estimation. This enables us to handle domains with sharp concavities, holes, and local pinches, such as polynomial sectors. Additionally, we introduce a data-driven selection rule based on the general ideas of Goldenshluger and Lepski. Our results demonstrate that the local polynomial estimators are minimax under a $L^2$ risk across a wide range of H\"older-type functional classes. In the adaptive case, we provide oracle inequalities and explicitly determine the convergence rate of our statistical procedure. Simulations on polynomial sectors show that our oracle estimates outperform those of the most popular alternative method, found in the sparr package for the R software. Our statistical procedure is implemented in an online R package which is readily accessible.