Nonparametric Density Estimation and Bandwidth Selection with B-spline bases: a Novel Galerkin Method

Abstract

We study a general and efficient nonparametric density estimation procedure for local bases, including B-splines, using a novel statistical Galerkin method, combined with basis duality theory. We provide an efficient cross-validation procedure to select the bandwidth, based on closed-form expressions in terms of the primal and dual basis. By utilizing a closed-form expression for the dual basis, we are able to compute the least-squares cross validation formula in closed-form, which allows us to effectively determine the bandwidth. The full computational procedure achieves optimal complexity, and is very accurate in comparisons with existing estimation procedures, including state-of-the-art kernel density estimators. Our theoretical results are supported by extensive numerical experiments, which demonstrate the efficiency and accuracy of the new methodology.