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

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. Our likelihood-based approach incorporates a regularization term addressing the roughness of the logarithm of density using a second-order differential operator. We establish the asymptotic convergence rate of the proposed density estimator in terms of the $L_2$ and $L_{\infty }$ norms under mild natural conditions, providing a solid theoretical foundation. The proposed method demonstrates superior efficiency and flexibility with enhanced smoothness and continuity across the domain compared to existing techniques. We validate our approach through comprehensive simulation studies and apply it to real-world motor vehicle theft data from Portland, Oregon, illustrating its practical advantages in data analysis on spatial domains.