Histogram-kernel Error and Its Application for Bin Width Selection in Histograms  

作　　者：Xiu-xiang Wang Jian-fang Zhang 

机构地区：[1]Department of Mathematics,Graduate University of Chinese Academy of Sciences,Beijing 100049,China [2]Coltege of Management,Graduate University of Chinese Academy of Sciences,Beijing 100190,China

出　　处：《Acta Mathematicae Applicatae Sinica》2012年第3期607-624,共18页应用数学学报（英文版）

基　　金：Supported by the National Natural Science Foundation of China (No. 70371018, 70572074)

摘　　要：Histogram and kernel estimators are usually regarded as the two main classical data-based nonparametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. 3ust as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott＇s optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.Histogram and kernel estimators are usually regarded as the two main classical data-based nonparametric tools to estimate the underlying density functions for some given data sets. In this paper we will integrate them and define a histogram-kernel error based on the integrated square error between histogram and binned kernel density estimator, and then exploit its asymptotic properties. 3ust as indicated in this paper, the histogram-kernel error only depends on the choice of bin width and the data for the given prior kernel densities. The asymptotic optimal bin width is derived by minimizing the mean histogram-kernel error. By comparing with Scott＇s optimal bin width formula for a histogram, a new method is proposed to construct the data-based histogram without knowledge of the underlying density function. Monte Carlo study is used to verify the usefulness of our method for different kinds of density functions and sample sizes.

关 键 词：HISTOGRAM binned kernel density estimator bin width histogram-kernel error integrated square error 

分 类 号：O174.41[理学—数学] TP391.41[理学—基础数学]