On the Rates of Asymptotic Normality for Bernstein Polynomial Estimators in a Triangular Array

It is well known that the empirical distribution function has superior properties as an estimator of the underlying distribution functionF. However, considering its jump discontinuities, the estimator is limited whenFis continuous. Mixtures of the binomial probabilities relying on Bernstein polynomials lead to good approximation properties for the resulting estimator ofF. In this paper, we establish the rates of (pointwise) asymptotic normality for Bernstein estimators by the Berry-Esseen Theorem in the case that the observations are in a triangular array. Particularly, the (asymptotic) absence of the boundary bias and the asymptotic behaviors of the variance are investigated. Besides, numerical simulations are presented to verify the validity of our main results.