多元统计分析及其应用  

Multivariate statistics and its applications

作　　者：李刚 梁家卷 潘建新 彭小令[4] 田国梁 Gang Li;Jiajuan Liang;Jianxin Pan;Xiaoling Peng;Guoliang Tian

机构地区：[1]Department of Biostatistics,Jonathan and Karin Fielding of Public Health,University of California at Los Angeles,Los Angeles,CA 90095-1772,USA [2]College of Business,University of New Haven,C T 06516,USA [3]Department of Mathematics,The University of Manchester M139PL,UK [4]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 [5]南方科技大学理学院统计与数据科学系,深圳51805

出　　处：《中国科学：数学》2020年第5期571-584,共14页Scientia Sinica：Mathematica

摘　　要：自20世纪50年代以来,多元统计的理论、方法及其应用受到了越来越广泛的关注.国内多元统计方向的研究始于20世纪30年代末至40年代初许宝騄在西南联合大学时期.现代大数据分析的需要使得古典多元统计方法不能完全有效地解决当前的实际问题.古典多元统计理论从20世纪70年代以来已经得到了快速发展,本文旨在对国内学者在推广古典多元统计理论及其应用方面的工作进行概述,主要包括:多元统计分析和广义多元统计、一般对称多元分布、增长曲线模型及其他方向.广义多元统计是正态假设下的传统统计方法论的推广.其目的是将传统的统计方法论,如参数估计、假设检验和统计模型等,推广到更大的多元分布族.这个分布族称为椭球等高分布族.一般对称多元分布构成一个更大的多元统计分布族.这个分布族包含了椭球等高分布族作为其特例.增长曲线模型包含了一类统计方法,它允许考虑个体内部及个体之间随着时间变化时的相关关系.异常观察点及影响观察点的辨别是增长曲线模型研究的一个重要方向.Multivariate statistics and its applications have received more and more attention since 1950s. Multivariate statistical research in China was initiated by Pao-Lu Hsu during the end of 1930s and the beginning of 1940s in the so-called "Southwest United University". Modern big data analysis makes classical multivariate statistical theory unable to accurately and effectively solve practical problems. The theory of generalized multivariate statistics has been developing very fast since 1970s. This paper aims to introduce the summary contributions that Chinese scholars have made in the development of generalized multivariate statistics and its applications in several aspects:(1) multivariate statistics and generalized multivariate statistics;(2) general symmetric multivariate distributions;and(3) growth curve modeling and miscellaneous directions. Generalized multivariate statistics is an extension to the traditional statistics under the normal assumption. It aims to generalize the traditional statistical methodologies like parametric estimation, hypothesis testing, and modeling to a much wider family of multivariate distributions that are called elliptically contoured distributions(ECDs). General symmetric multivariate distributions form an even wider class of multivariate probability distributions that includes the ECDs as its special case. Growth curve modeling(GCM) includes statistical methods that allow for consideration of inter-individual variability in intra-individual patterns of chance over time. Outlier detection and identification of influential observations are important topics in the area of GCM.

关 键 词：COPULA l1-模对称分布 球对称分布 随机表示 椭球等高分布 占有问题 增长曲线模型 左球矩阵分布 

分 类 号：O212[理学—概率论与数理统计]