On improving convergence rate of Bernstein polynomial density estimator
On improving convergence rate of Bernstein polynomial density estimator
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DOI：
10.1080/10485252.2013.827195
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发表时间：
2014-01-02
影响因子：
1.2
通讯作者：
Kakizawa, Yoshihide
中科院分区：
文献类型：
--
作者：
Igarashi, Gaku;Kakizawa, Yoshihide
本文讨论了伯恩斯坦估计[Vitale，R.A.（1975），“一个伯恩斯坦多项式方法密度函数估计”，统计推断和相关主题，编辑。Puri，2，纽约：学术出版社，pp. 87-99]来估计具有支持[0，1]的密度。本文的主要贡献之一是乘法偏差校正的应用[特雷尔，G.R.，和斯科特，D. W.（1980），“On Improving Convergence Rates for Nonnegative Kernel Density Estimators”，The Annals of Statistics，8，1160-1163]，其最初是针对标准核估计量开发的。此外，还严格研究了重正化乘性偏差校正的伯恩斯坦估计.所得偏差校正伯恩斯坦估计量以及加性偏差校正伯恩斯坦估计量的内部均方误差（MSE）和平均积分均方误差[Leblanc，A.（2010），'A Bias-reduced Approach to Density Estimation Using伯恩斯坦Polynomials'，Journal of Nonparametric Statistics，22，459-475]当底层密度具有四阶导数时，证明为O（n−8/9），其中是样本大小。本文还讨论了在边界附近均方误差为O（n−8/9）的条件。最后，基于模拟和真实的数据集的数值研究。
This paper is concerned with the Bernstein estimator [Vitale, R.A. (1975), ‘A Bernstein Polynomial Approach to Density Function Estimation’, inStatistical Inference and Related Topics, ed. M.L. Puri, 2, New York: Academic Press, pp. 87–99] to estimate a density with support [0, 1]. One of the major contributions of this paper is an application of a multiplicative bias correction [Terrell, G.R., and Scott, D.W. (1980), ‘On Improving Convergence Rates for Nonnegative Kernel Density Estimators’,The Annals of Statistics, 8, 1160–1163], which was originally developed for the standard kernel estimator. Moreover, the renormalised multiplicative bias corrected Bernstein estimator is studied rigorously. The mean squared error (MSE) in the interior and mean integrated squared error of the resulting bias corrected Bernstein estimators as well as the additive bias corrected Bernstein estimator [Leblanc, A. (2010), ‘A Bias-reduced Approach to Density Estimation Using Bernstein Polynomials’,Journal of Nonparametric Statistics, 22, 459–475] are shown to beO(n−8/9) when the underlying density has a fourth-order derivative, wherenis the sample size. The condition under which the MSE near the boundary isO(n−8/9) is also discussed. Finally, numerical studies based on both simulated and real data sets are presented.