Bernstein polynomial probability density estimation
Bernstein polynomial probability density estimation
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DOI：
10.1080/1048525042000191486
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发表时间：
2004-10-01
影响因子：
1.2
通讯作者：
Kakizawa, Y
中科院分区：
文献类型：
--
作者：
Kakizawa, Y
我们考虑应用 Bernstein 多项式来估计支持 [0, 1 ] 的密度函数。本文提出的两类估计量被解释为 in 或 m + 1 点处的（边界）核估计量的线性组合，其系数是二项式分布 Bin(m - 1, x) 或 Bin(m, x) 的概率，x 是进行密度估计的位置。结果表明，我们的估计量没有边界偏差，并且均方误差的收敛速度达到 n(-4/5)。许多估计量仍然是非负的，这与 Chen 的变量相当[Chen, S. X. (1999)。密度函数的 Beta 核估计器。计算统计与数据分析，31, 131-145.]（边界）β 核估计器。我们的第一类基于带有边界修改的统一内核的估计器包括 Vitale 的估计器 [Vitale, R. A. (1975)。密度函数估计的伯恩斯坦多项式方法。见：Puri, M. L.（编辑），统计推断和相关主题，卷。 2.学术出版社，纽约，第87-99页。]作为一个特殊的笼子，这个子类中的一些估计量在渐近均方误差方面优于陈的第一个估计量。此外，利用 Bernstein 多项式方法提出了三个估计量，它们不仅优于 Vitale 的估计量，而且与 Chen 的第二估计量相当。
We consider an application of Bernstein polynomials for estimating a density function with support [0, 1 ]. Two classes of estimators proposed in this article are interpreted as a linear combination of (boundary) kernel estimators at in or m + 1 points, whose coefficients are probabilities of the binomial distribution Bin(m - 1, x) or Bin(m, x), x being the position where the density estimation is made. It is shown that our estimators are free of boundary bias and achieve the convergence rate of n(-4/5) for the mean integrated squared error. Many estimators remain nonnegative, which are comparable with Chen's variable [Chen, S. X. (1999). Beta kernel estimators for density functions. Computational Statistics & Data Analysis, 31, 131-145.] (boundary) beta kernel estimators. Our first class of stimators based on the uniform kernel with boundary modification includes Vitale's estimator [Vitale, R. A. (1975). A Bernstein polynomial approach to density function estimation. In: Puri, M. L. (Ed.), Statistical Inference and Related Topics, Vol. 2. Academic Press, New York, pp. 87-99.] as a special cage, and some estimators in this subclass are superior to Chen's first estimator in terms of the asymptotic mean integrated squared error. Further, three estimators that are not only superior to Vitale's estimator but also equivalent to Chen's second estimator are proposed by using the Bernstein polynomial approach.