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.

众所周知，经验分布函数具有作为基础分布函数F的估计量的优良特性。但是，考虑到其跳跃不连续性，当F连续时，估计量会受到限制。依赖伯恩斯坦多项式的二项式概率的混合导致F的最终估计量具有良好的近似性质。在本文中，当观测值呈三角形排列时，我们通过Berry-Esseen定理建立Bernstein估计量的（逐点）渐近正态率。特别是，研究了边界偏差的（渐近）不存在和方差的渐近行为。此外，通过数值模拟来验证我们主要结果的有效性。