A functional data analysis approach to the estimation of densities over complex regions

In this work we propose a nonparametric method for density estimation over two-dimensional domains. Following a functional data analysis approach, we consider a penalized likelihood estimator, with a roughness penalty based on a differential operator. This approach allows for the estimation of densities on any planar domain, including those with complex boundaries or interior holes. We develop an estimation procedure based on finite elements. Thanks to the use of this numerical technique, the proposed method has great flexibility and high computational efficiency.