Convergence Rates for Uniform B-Spline Density Estimators Part I: One Dimension

A B-spline nonparametric density estimator with uniform knots, convenient for solving problems in computer graphics, was discussed by Gehringer and Redner in 1992 [ Comm. Stat. Simulation Methods, 21 (1992) pp. 849--878]. These ideas were later extended to density function estimation on metric spaces using partitions of unity in Redner and Gehringer in 1994 [Comm. Stat. Theory Methods, 23 (1994) pp. 2059--2076]. In the current paper we return to the uniform B-spline density estimator in one dimension. We show, under very natural assumptions, that the B-spline density estimator and all of its nontrivial derivatives converge in mean integrated squared error. Asymptotic rates of convergence are determined for the density estimate and its derivatives.