论文标题
关于伯恩斯坦估计量在单纯形上的边界特性
On the boundary properties of Bernstein estimators on the simplex
论文作者
论文摘要
在本文中，我们研究了伯恩斯坦估计量的渐近特性（偏差，方差，平方误差），用于累积分布函数和密度函数，附近和密度函数在$ d $ d $ dimemensional单纯的边界上。我们的结果概括了LeBlanc（2012）发现的那些，后者对案件进行了$ d = 1 $的处理，并补充了单纯形内部的Ouimet（2021）的结果。由于$ d $维单纯胶的“边缘”的尺寸从$ 0 $（顶点）到$ d-1 $（1 $）的尺寸，我们的内核功能是多项式的，因此，偏见，差异和平方误差的渐近表达并非直接范围，因为它们几乎可以估计，因此，这些产品的估计是，这些产品的概述是对产品的估计，而这些产品的概述是构成产品的范围。伯恩斯坦估计器或不对称内核估计器。这一点使数学分析更加有趣。
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.
