Finite element thin plate splines in density estimation

Abstract

The problem of estimating probability density functions differs significantly from functional estimation in which a response variable is present and has for this reason has been dealt with by substantially different methods. We demonstrate here that it is possible to apply spline-type functionals to the problem of density estimation for large data sets. The resulting estimators may be regarded as kernel methods, but may also be applied to inexact or aggregated data. They can be seen to have moments matching the empirical moments of the data up to the degree of smoothness of the function. Finally, we will show that these functions may be naturally approximated by a finite element method and that doing so will make the method scalable.