The Random Bernstein Polynomial Smoothing Via ABC Method

Abstract

In recent years, many statistical inference problems have been solved by using Markov Chain Monte Carlo (MCMC) techniques. However, it is necessary to derivate the analytical form for the likelihood function. Although the level of computing has increased steadily, there is a limitation caused by the difficulty or the misunderstanding of how computing the likelihood function. The Approximate Bayesian Computation (ABC) method dispenses the use of the likelihood function by simulating candidates of posterior distributions and using an algorithm to accept or reject the proposed candidates. This work presents an alternative nonparametric estimation method of smoothing empirical distributions with random Bernstein polynomials via ABC method. The Bernstein prior is obtained by rewriting the Bernstein polynomial in terms of k mixtures / m mixtures of beta densities and mixing weights. A study of simulation and a real example are presented to illustrate the method proposed.