Nonparametric Density Estimation for Data Scattered on Irregular Spatial Domains: A Likelihood-Based Approach Using Bivariate Penalized Spline Smoothing

Abstract: 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.