Application of iterated Bernstein operators to distribution function and density approximation

We propose a density approximation method based on Bernstein polynomials, consisting in superseding the classical Bernstein operator by a convenient number I∗ of iterates of a closely related operator. We mainly tackle two difficulties met in processing real data, sampled on some mesh XN . The first one consists in determining an optimal sub-mesh XK∗ , in order that the operator associated with XK∗ can be considered as an authentic Bernstein operator (necessarily associated with a uniform mesh). The second one consists in optimizing I in order that the approximated density is bona fide (positive and integrates to one). The proposed method is tested on two benchmarks in Density Estimation, and on a grain-size curve.