Abstract
It is well known that the empirical distribution function has superior properties as an estimator of the underlying distribution function F. However, considering its jump discontinuities, the estimator is limited when F is continuous. Mixtures of the binomial probabilities relying on Bernstein polynomials lead to good approximation properties for the resulting estimator of F. 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.
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