论文标题
伯恩斯坦估计量在单纯形上的渐近特性
Asymptotic properties of Bernstein estimators on the simplex
论文作者
论文摘要
伯恩斯坦估计器众所周知，可以避免传统核估计器的边界偏见问题。这些估计量的理论特性已在紧凑的间隔和超启发上进行了广泛的研究，但绝不是单纯的，除了$ d = 2 $时，除了Tenbusch（1994）中密度估计器的平均平方误差。单纯形是一个重要的情况，因为它是组成数据的自然领域。在本文中，我们努力证明伯恩斯坦在$ d $ d $ dimensional-dimensional-dimensional-dimensional-dimensional-dimensional-dimensional-dimensional-dy-dimensional-dimensional累积的分布功能和密度功能的伯恩斯坦估计器中证明了几个渐近结果（偏差，方差，平均平方误差（MSE），平均集成平方误差（MSE），渐变态性，均匀的强度一致性）。我们的结果概括了LeBlanc（2012）和Babu等人的结果。 （2002年），他对案件进行了$ d = 1 $的处理，并大大扩展了Tenbusch（1994）中发现的案例。特别是，我们的MSE和MISE收敛速度是最佳的。
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 (2012) 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.
