Efficient and robust density estimation using Bernstein type polynomials

A method of parameterising and smoothing the unknown underlying distributions using Bernstein type polynomials with positive coefficients is proposed, verified and investigated. Any distribution with bounded and smooth enough density can be approximated by the proposed model which turns out to be a mixture of the beta distributions, beta, , for some optimal degree m. A simple change-point estimating method for choosing the optimal degree m of the approximate model is presented. The proposed method gives a maximum likelihood density estimate which is consistent in distance at a nearly parametric rate under some conditions. Simulation study shows that one can benefit from both the smoothness and the efficiency by using the proposed method which can also be used to estimate some population parameters such as the mean. The proposed methods are applied to three data sets of different types.