Distribution and density estimation based on variation-diminishing spline approximation

The estimation of distribution functions is a fundamental problem in the fields of statistics and machine learning. In this paper, based on Schoenberg’s variation-diminishing spline approximation, we propose an efficient nonparametric method for estimating distribution and density functions with bounded support. The preservation of the monotonicity of the distribution function and the positivity of the density function are guaranteed. Both methodology and asymptotic properties are established. We demonstrate theoretically and numerically that the smooth Schoenberg’s estimator can outperform the empirical cumulative distribution function. Several simulated examples and real data example are given to illustrate the efficiency and performance of our method. In the simulation study, the approach achieves very competitive performance with the kernel and the Bernstein polynomial estimators.