Nonparametric density estimation for scattered spatial data on irregular domains: a likelihood-based approach using bivariate penalised spline smoothing

Accurate density estimation of spatial data is crucial for informed decision-making and modelling across various fields. This paper presents a novel nonparametric density estimation procedure using bivariate penalised spline smoothing over triangulation for data scattered across irregular domains. Our likelihood-based approach incorporates a regularisation term addressing the roughness of the density logarithm using a second-order differential operator. We establish the asymptotic convergence rate of the proposed density estimator in terms of the $ L_2 $ L2 and $ L_{\infty } $ L∞ 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 spatial data analysis.