统计分布的代表点集及其应用  

Sets of representative points of statistical distributions and their applications

作　　者：方开泰[1,2] 贺平[1] 杨骏 Kaitai Fang;Ping He;Jun Yang

机构地区：[1]北京师范大学-香港浸会大学联合国际学院理工科技学部,珠海519087 [2]中国科学院数学与系统科学研究院应用数学研究所,北京100190

出　　处：《中国科学：数学》2020年第9期1149-1168,共20页Scientia Sinica：Mathematica

基　　金：珠海市优势学科基金(批准号:R1050);北京师范大学-香港浸会大学联合国际学院校内科研基金(批准号:R201712,R201810,R201912和R202010)资助项目。

摘　　要：用一个离散统计分布来近似一个连续的统计分布(一维或多维)一直是统计学研究的核心内容.显然这个离散统计分布的支撑点集必须有代表性,故称它们为代表点集,或简称代表点.选择代表点可以有不同的考虑,本文回顾并比较4类近似离散统计分布:随机样本(独立同分布)、修改的Monte Carlo方法、数论方法的样本(伪Monte Carlo方法)及在最小平方误准则下的代表点集和相应的统计分布.其中修改的Monte Carlo方法是本文新提出的.本文比较4类方法在密度估计和重采样的统计推断中的表现,其中有一类是改进的自助法.本文对最小平方误准则下的代表点的性质和数值算法进行了详细回顾,并且得到一些新结果,例如,随机样本的最小平方误准则的统计分布、椭球等高分布代表点的几何结构以及椭球等高分布代表点和主成分的关系.An important issue of how to use a discrete distribution to approximate a given continuous statistical distribution has been studied in statistics.Obviously,the support points of this discrete distribution must have a good representative in a certain sense.The set of these support points are called representative points (RPs).There are different considerations for representation.This paper reviews four approaches:Monte Carlo (i.i.d.),revised Monte Carlo,number-theoretic methods,and the mean squared error criterion.The revised Monte Carlo method is new.We compare the performance of resampling by these four methods in density estimation and statistical inference,one of which is a revised bootstrap method.The paper pays more attention to the properties of MSE (mean squared error) representative points and algorithms for their generation.Some new results are obtained,for example,the distribution of MSE,the geometric pattern of RPs of elliptical distributions and relationships between the RPs and principal components.

关 键 词：统计分布代表点 伪Monte Carlo方法 统计推断 正态分布 椭球等高分布 主成分和主成分点 

分 类 号：O212[理学—概率论与数理统计]