On the rates of asymptotic normality for Bernstein density estimators in a triangular array

Abstract

In this paper, we consider the Bernstein polynomial of the empirical density fn under a triangular sample, which we denote by fˆm,n. For the recentered and normalized statistic n1/2m−1/4(fˆm,n(x)−Efˆm,n(x)), where x is defined on the interval (0,1), the convergence rate to the normal distribution is established by the Berry-Esseen theorem. We also obtain asymptotic expressions for the pointwise bias and variance. In addition, numerical simulations are presented to verify the validity of our main results.