Optimal stochastic Bernstein polynomials in Ditzian–Totik type modulus of smoothness

We introduce a family of symmetric stochastic Bernstein polynomials based on order statistics, and establish the order of convergence in probability in terms of the second order Ditzian–Totik type modulus of smoothness on the interval [0,1], which epitomizes an optimal pointwise error estimate for the classical Bernstein polynomial approximation. Monte Carlo simulation results (presented at the end of the article) show that this new approximation scheme is efficient and robust.