A general and efficient nonparametric density estimation procedure for local bases, including B-splines, is proposed, which employs a novel statistical Galerkin method combined with basis duality theory. To select the bandwidth, an efficient cross-validation procedure is introduced, based on closed-form expressions in terms of the primal and dual B-spline basis. By utilizing a closed-form expression for the dual basis, the least-squares cross validation formula is calculated in closed-form, enabling an efficient estimation of the optimal bandwidth. The full computational procedure achieves optimal complexity, and is very accurate in comparison with existing estimation procedures, including state-of-the-art kernel density estimators. The presented theoretical results are supported by extensive numerical experiments, which demonstrate the efficiency and accuracy of the new methodology. This new approach provides a complete and optimally efficient framework for density estimation with a B-spline basis, based on simple and elegant closed-form estimators with theoretical convergence results that are substantiated in numerical experiments.

提出了一种通用的，有效的局部参数非参数密度估计程序，包括B样条曲线，它结合了新颖的统​​计Galerkin方法和基础对偶理论。为了选择带宽，基于原始形式和双重B样条的封闭形式表达式，引入了有效的交叉验证过程。通过使用对偶的封闭式表达式，可以以封闭式计算最小二乘交叉验证公式，从而可以高效估计最佳带宽。完整的计算过程可实现最佳的复杂性，并且与现有的估算过程（包括最新的内核密度估算器）相比非常准确。提出的理论结果得到广泛的数值实验的支持，证明了新方法的效率和准确性。这种新方法基于简单且优雅的闭式估计量，并在数值实验中得到了理论上的收敛结果，从而为基于B样条的密度估计提供了一个完整且最优的高效框架。