Nonparametric Density Estimation Using Flexible NURBS Modeling

Nonparametric methods for density estimation provide flexible tools for modeling data distributions without assuming a specific parametric form. In this study, we propose a novel approach based on Non-Uniform Rational B-Splines (NURBS) to estimate probability density functions in a fully data-driven manner. The estimator is constructed by optimizing a set of control points and associated weights under constraints that ensure non-negativity and unit integral, guaranteeing that the resulting function is a valid density. Unlike classical polynomial-based estimators, the NURBS framework offers enhanced flexibility by accommodating non-uniform knot vectors and rational weighting, allowing for better adaptation to sharp features and multimodal structures. The performance of the proposed estimator is examined through simulations involving a wide range of distributional shapes, and its practical performance is demonstrated using real-world datasets. Comparative results indicate that the NURBS-based estimator provides competitive or superior accuracy compared to traditional Bernstein and Bézier-based alternatives, especially in complex distributional settings.