Bernstein polynomial probability density estimation

We consider an application of Bernstein polynomials for estimating a density function with support [0, 1]. Two classes of estimators proposed in this article are interpreted as a linear combination of (boundary) kernel estimators at m or m + 1 points, whose coefficients are probabilities of the binomial distribution Bin (m − 1, x) or Bin (m, x), x being the position where the density estimation is made. It is shown that our estimators are free of boundary bias and achieve the convergence rate of n −4/5 for the mean integrated squared error. Many estimators remain nonnegative, which are comparable with Chen's variable [Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics & Data Analysis, 31, 131–145.] (boundary) beta kernel estimators. Our first class of estimators based on the uniform kernel with boundary modification includes Vitale's estimator [Vitale, R. A. (1975). A Bernstein polynomial approach to density function estimation. In: Puri, M. L. (Ed.), Statistical Inference and Related Topics, Vol. 2. Academic Press, New York, pp. 87–99.] as a special case, and some estimators in this subclass are superior to Chen's first estimator in terms of the asymptotic mean integrated squared error. Further, three estimators that are not only superior to Vitale's estimator but also equivalent to Chen's second estimator are proposed by using the Bernstein polynomial approach.