A note on generalized Bernstein polynomial density estimators

Abstract

We propose a rescaled generalized Bernstein polynomial for approximating any continuous function defined on the closed interval [ 0 , Δ ] . 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 / Δ ) , depending on the location x ∈ [ 0 , Δ ] 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 ∈ 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.