多输出分位数回归估计中若干问题的研究

In statistics, the quantile regression method has been widely used in many practical fields due to its powerful ability to describe the features of data, and induce some robust estimations, as an alternative to the classical least squares method thought. The current quantile regression method can only be applied in situations when the response is single-outputting, nevertheless. In the setting of multiple-output, the existing quantile regression method can’t be directly utilized. Observe that the phenomenon of multiple-output commonly exists in practice, and the related researcher under such a new scenario is quite limited up to the present. There are still a lot of works worthy of further considerations. To this end, this project is planning to: (1) study the robustness properties, such as the finite sample/limiting breakdown point of the multivariate median; (2) consider the quantile regression estimation in the multiple-output linear model, multiple-output nonparametric model and multiple-output semi-parametric model, and investigate their theoretical properties and finite sample performance; (3) explore the computing issue of the developed methods, and construct some feasible and fast (both exact and approximate) algorithms, as well as the corresponding package, to facilitate their practical applications. This project can be classified as researches of both methogology 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 multiple output quantile regression, but also bring benefit to the computation of these procedures, which in turn probably improve the usability of the multiple output quantile regression and their related methods to a great extend in the practical data analysis.

在统计学中，分位数回归方法因其对数据的描述功能非常强大、且可以诱导出某些稳健估计而常被作为经典最小二乘方法的有益补充，应用于在实际数据分析的诸多领域。然而，现有分位数回归方法主要针对响应变量为单输出的情形。在多输出情形下，现有分位数回归方法无法直接应用。考虑到多输出现象在现实中普遍存在，相关情形下分位数回归估计的研究为时尚浅，仍有大量问题值得深入探讨。因此本项目拟：(1)考察多输出分位数回归在位置情形下对应的多元中位数的崩溃点性质；(2)研究多输出线性模型、非参数模型和半参数模型的分位数回归估计，并考察它们的理论性质和有限样本表现；(3)深入开展多输出分位数回归方法的精确与近似计算算法研究，编写相应程序包，以方便它们今后可能的实证应用。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。它的推进可望能为多输出分位数回归方法带来新发展，为其计算注入新活力，进而促进它们的实际应用。

在统计学中，分位数及由其推广而来的分位数回归方法因其对数据的描述功能非常强大、且可以诱导出某些稳健估计而常被应用于在实际数据分析的诸多领域，它们一起构成了稳健统计的核心内容。然而，现有的分位数及其相关回归情形下的分位数回归方法主要针对响应变量为单输出的情形。在多元数据情形下，如何构架相应的多元分位数及相关的多输出分位数回归估计是近年来多元稳健统计关心的重要问题。相关研究虽然已有见诸于文献，如Tukey,W.J.1975年从统计深度函数的角度提出了Tukey多元分位数(也常被称为深度域)等，但仍有大量问题值得深入探讨。因此本项目对上述相关内容的理论性质、计算方法开展了深入的研究。具体内容如下：.1、位置情形下多元中位数的稳健性质.(1) 在有限样本情形下，推导D(x,Fn关于x的最小上界值；.(2) 在有限样本情形下，求取多元中位数的精确崩溃点值；.(3) 在分布函数为Halfspace对称的假定下，推导多元中位数的渐近崩溃点；.2、回归情形下多输出分位数回归估计及其理论性质.(1) 推导线性模型多输出分位数回归估计的稳健性质；.(2) 推导非参数模型多输出分位数回归估计的大样本；.(4) 推导半参数模型多输出分位数回归估计的大样本；.3、多输出分位数回归估计的计算.(1) 考察多元分位数的快速计算算法；.(2) 考察非参数模型、半参数模型下多输出分位数回归估计的计算算法；.(3) 编写基于Matlab的多输出分位数回归统计软件包。.在本项目基金的资助下，项目组业已完成了项目预设的绝大部分研究内容，完整地解决了Tukey多元分位数的快速、精确计算算法设计、Tukey中位数有限样本崩溃点精确表达式的求取等本领域公开问题。成果得到了同行专家的认可，并陆续发表在《中国科学》、《Journal of Computational and Graphical Statistics》等国内外统计学知名刊物上，论文共计13篇，其中申请人一作或者通信作者的论文9篇。.