Iterated Bernstein operators for distribution function and density estimation: Balancing between the number of iterations and the polynomial degree

Despite its slow convergence, the use of the Bernstein polynomial approximation is becoming more frequent in Statistics, especially for density estimation of compactly supported probability distributions. This is due to its numerous attractive properties, from both an approximation (uniform shape-preserving approximation, etc.) and a statistical (bona fide estimation, low boundary bias, etc.) point of view. An original method for estimating distribution functions and densities with Bernstein polynomials is proposed, which takes advantage of results about the eigenstructure of the Bernstein operator to refine a convergence acceleration method. Furthermore, an original data-driven method for choosing the degree of the polynomial is worked out. The method is successfully applied to two data-sets which are important benchmarks in the field of Density Estimation.