Nonparametric density estimation over complicated domains

We propose a nonparametric method for density estimation over (possibly complicated) spatial domains. The method combines a likelihood approach with a regularization based on a differential operator. We demonstrate the good inferential properties of the method. Moreover, we develop an estimation procedure based on advanced numerical techniques, and in particular making use of finite elements. This ensures high computational efficiency and enables great flexibility. The proposed method efficiently deals with data scattered over regions having complicated shapes, featuring complex boundaries, sharp concavities or holes. Moreover, it captures very well complicated signals having multiple modes with different directions and intensities of anisotropy. We show the comparative advantages of the proposed approach over state of the art methods, in simulation studies and in an application to the study of criminality in the city of Portland, Oregon.