In this paper, we study the asymptotic properties (bias, variance, mean squared error) of Bernstein estimators for cumulative distribution functions and density functions near and on the boundary of the $d$-dimensional simplex. Our results generalize those found by Leblanc (2012), who treated the case $d=1$, and complement the results from Ouimet (2021) in the interior of the simplex. Since the "edges" of the $d$-dimensional simplex have dimensions going from $0$ (vertices) up to $d - 1$ (facets) and our kernel function is multinomial, the asymptotic expressions for the bias, variance and mean squared error are not straightforward extensions of one-dimensional asymptotics as they would be for product-type estimators studied by almost all past authors in the context of Bernstein estimators or asymmetric kernel estimators. This point makes the mathematical analysis much more interesting.

翻译：在本文中,我们研究了Bernstein测算器的累积分布函数和密度函数在美元维度简单度的附近和边界的累积分布函数和密度函数的无症状属性(比例、差异、平均平方误差)。我们的结果概括了Leblanc(2012年)发现的结果,后者处理了案件 $d=1美元,并补充了Oumit (2021年) 在简单x内部的 Oumit (2021年) 的结果。由于美元维度简单度的“边缘”的维度从0美元(垂直值)到1美元(脸部),而我们的内核函数是多数值的,偏差、差异和平均平方差误差的无症状表达方式并不是对几乎过去所有作者在Bernstein测算器或不对称内核测算器中研究的产品型测算器的单维度器的直截面延伸。这让数学分析更加有趣。