Nonparametric density estimation with nonuniform B-spline bases

Abstract

Recently, Kirkby et al. introduced a general and efficient nonparametric density estimation procedure for local bases that is based on the statistical Galerkin method. However, the resulting density estimators using the statistical Galerkin method are not guaranteed to be proper, i.e., the density functions are not ensured to be nonnegative and normalized. To address this issue, we present a constrained Galerkin method with B-spline functions for nonparametric density estimation. Furthermore, we also provide a procedure to generate the optimal knot vectors for density estimations with nonuniform B-spline bases, which are based on the cumulative distribution of the feature function. Extensive numerical experiments are conducted to demonstrate the efficiency and accuracy of our framework.