Accurately estimating data density is crucial for making informed decisions and modeling in various fields. This paper presents a novel nonparametric density estimation procedure that utilizes bivariate penalized spline smoothing over triangulation for data scattered over irregular spatial domains. The approach is likelihood-based with a regularization term that addresses the roughness of the logarithm of density based on a second-order differential operator. The proposed method offers greater efficiency and flexibility in estimating density over complex domains and has been theoretically supported by establishing the asymptotic convergence rate under mild natural conditions. Through extensive simulation studies and a real-world application that analyzes motor vehicle theft data from Portland City, Oregon, we demonstrate the advantages of the proposed method over existing techniques detailed in the literature.

翻译：精确估计数据密度对于各领域中的决策制定与建模至关重要。本文提出一种新颖的非参数密度估计方法，该方法利用基于三角剖分的二元惩罚样条平滑技术来处理不规则空间域上的散乱数据。该方法基于似然，并引入一个正则化项，该正则化项基于二阶微分算子来处理密度对数的粗糙度。所提方法在复杂区域上的密度估计中展现出更高的效率与灵活性，并在温和的自然条件下通过建立渐近收敛速率获得了理论支持。通过大量的模拟研究以及一个分析美国俄勒冈州波特兰市机动车盗窃数据的实际应用，我们证明了所提方法相较于文献中详述的现有技术所具有的优势。