统计深度函数的若干推广及其计算方法研究

项目名称： 统计深度函数的若干推广及其计算方法研究

项目编号： No.11461029

项目类型： 地区科学基金项目

立项/批准年度： 2015

项目学科： 数理科学和化学

项目作者： 刘小惠

作者单位： 江西财经大学

项目金额： 36万元

中文摘要： 在一元统计学中，次序统计量应用广泛。它同极大似然法、矩法及最小二乘法等地位类似，是众多统计推断方法的构造基础。在多元情形下，统计深度函数因其能为数据提供某种由中心而外的次序而被广泛用于诱导各种性质优良的多元方法，起着与次序统计量相似的作用。本项目拟：(1)针对现有深度函数在数据维数相对样本容量较大时性能表现较差的客观实际，构建高维深度函数，并考察与之相关的理论性质与实际应用；(2)建立回归情形下可对回归参数排序的投影回归深度函数，给出一般回归深度函数的公理化体系，研究其诱导估计的理论性质与置信域构造方法，相关构造方法也可用于现有统计深度函数的诱导估计上；(3)深入开展统计深度函数及其推广形式精确与近似算法的研究，编写相应计算程序包。本项目属理论和方法研究，所研究问题是对当今多元统计的丰富和发展。它的推进可望能为现有统计深度函数带来新发展，为其计算注入新活力，进而促进相关方法的实际应用。

中文关键词： 非参数统计；经验似然；多元统计分析；统计推断

英文摘要： In univariate statistics, the order statistics are widely used. They lay the foundation for many statistical inference procedures, as did by the maximum likelihood method, the moment method and the least square method, etc.. In the multivariate setting, the statistical depth functions play a similar role of order statistics and are widely utilized to induce various desirable multivariate methods, because they are capable to provide a center-outward ordering for the multivariate observations. In this project, we are interested to: (1) extend the current statistical depth functions to versions that can order observations, to improve their poor performances when the dimension may be very high relative to the sample size, and investigagte the related theoretical properties and pratical applications; (2) in the regression setting, propose some new projection regression depths, develop a general notion of regression depth, and study the theoretical properties and the confidence region construction of the associated estimators,which includes the confidence region construction of the existing depth function based estimators as a particular case; (3) study the computing issue of statistical depth functions and the related extensions, and construct some feasible and fast (both exact and approximate) algorithms, as well as the corresponding package. This project can be classed as researches of both theory and method. It can be considered as a development of the current multivariate statistics. Ideally, the researches in this project are expected to not only be helpful in enriching the statistical depth functions, but also bring benefit to the computation of these depth-induced procedures, which in turn probably improve the usability of the depth functions and their related methods to a great extend in the practical data analysis.

英文关键词： Nonparametric statistics;Empirical likelihood;Multivariate data analysis;Statistical inference