Bernstein estimators are well-known to avoid the boundary bias problem of traditional kernel estimators. The theoretical properties of these estimators have been studied extensively on compact intervals and hypercubes, but never on the simplex, except for the mean squared error of the density estimator in Tenbusch (1994) when $d=2$. The simplex is an important case as it is the natural domain of compositional data. In this paper, we make an effort to prove several asymptotic results (bias, variance, mean squared error (MSE), mean integrated squared error (MISE), asymptotic normality, uniform strong consistency) for Bernstein estimators of cumulative distribution functions and density functions on the $d$-dimensional simplex. Our results generalize the ones in Leblanc (2012a) and Babu et al. (2002), who treated the case $d=1$, and significantly extend those found in Tenbusch (1994). In particular, our rates of convergence for the MSE and MISE are optimal.

众所周知，伯恩斯坦估计器可以避免传统核估计器的边界偏差问题。这些估计量的理论性质已经在紧致区间和超立方体上得到了广泛的研究，但从未在单纯形上进行过研究，除了 Tenbusch (1994) 中密度估计量的均方误差，当$d=2$. 单纯形是一个重要的例子，因为它是组合数据的自然域。在本文中，我们努力证明累积分布函数和密度函数的 Bernstein 估计量的几个渐近结果（偏差、方差、均方误差 (MSE)、平均积分平方误差 (MISE)、渐近正态性、均匀强一致性）在$d$维单纯形。我们的结果概括了 Leblanc (2012a) 和 Babu 等人的结果。(2002)，谁处理了这个案子$d=1$，并显着扩展了 Tenbusch (1994) 中发现的那些。特别是，我们对 MSE 和 MISE 的收敛速度是最佳的。