A note on generalized Bernstein polynomial density estimators

We propose a rescaled generalized Bernstein polynomial for approximating any continuous function defined on the closed interval [0, Delta]. Using this polynomial which is of degree m - 1 and depends on the additional parameter s(m), we consider the nonparametric density estimation for two contexts. One is that of a spectral density function of a real-valued stationary process, and the other is that of a probability density function with support [0, 1]. Our density estimators can be interpreted as a convex combination of the uniform kernel density estimators at m points, whose coefficients are probabilities of the binomial random variable with parameters (m - 1, x/Delta), depending on the location x is an element of [0, Delta] where the density estimation is made. We examine in detail the asymptotic bias, variance and mean integrated squared error for a class of our density estimators under the framework where m is an element of N tends to infinity in some way as the sample size tends to infinity. Using a specific data set, we also include a numerical comparison between our density estimators and the Bernstein-Kantorovich polynomial density estimator obtained through the cross-validation method. (C) 2010 Elsevier BM. All rights reserved.