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.

翻译：本文提出了一种新颖的方法，用于在已知任意维度的域上，利用非参数局部多项式估计量对多元密度函数进行逐点估计。我们的方法具有高度灵活性，既适用于简单域（如开连通集），也适用于更复杂的域（如非星形于估计点周围的域）。这使得我们能够处理具有尖凹性、空洞和局部收缩的域，例如多项式扇形域。此外，我们基于Goldenshluger和Lepski的通用思想，引入了一种数据驱动的选择规则。结果表明，在一系列Hölder型函数类中，局部多项式估计量在$L^2$风险下达到极小极大最优性。在自适应情形下，我们提供了奥拉克不等式，并明确确定了该统计过程的收敛速度。在多项式扇形域上的模拟显示，我们的奥拉克估计优于最流行的替代方法（即R软件中sparr包提供的方法）。该统计过程已在在线R包中实现，并可供直接使用。