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,Δ][0,Δ]. Using this polynomial which is of degree m−1m−1 and depends on the additional parameter smsm, 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][0,1]. Our density estimators can be interpreted as a convex combination of the uniform kernel density estimators at mm points, whose coefficients are probabilities of the binomial random variable with parameters (m−1,x/Δ)(m−1,x/Δ), depending on the location x∈[0,Δ]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.