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统计学中的一些矩阵理论及其相关应用

更新时间 2009-9-17 13:41:50 点击数:

统计学中的一些矩阵理论及其相关应用
Some Matrix Theory in Statistics and Its Application
【中文摘要】 矩阵理论在现代统计学的许多分支都有广泛的应用,成为统计学中不可缺少的工具。同时统计学中又提出了许多新的有关矩阵论的课题,刺激了矩阵论的发展。本文给出了矩阵论中与统计学密切相关的几个方面,讨论了这些结果的统计应用,特别是在线性模型参数估计和多元分析中的应用,这些结果都是在原有理论的基础上的推广。本文取得的主要结果如下:1.第三章主要讨论矩阵偏序及其在线性模型比较中的应用。首先,简单介绍了估计和模型比较的基础知识,然后在本章第三节利用矩阵偏序理论比较了广义岭估计与LS估计。在均方误差准则下,一些文献讨论了岭估计优于LS估计的问题,本章在此基础上,利用Lowner偏序讨论了广义岭估计相对于LS估计的优良性质,推广了已有的结论。2.第四章主要介绍了两类矩阵不等式及其在线性统计中的应用。首先,在本章第二节介绍了Kantorovich不等式矩阵形式及其统计应用。Kantorovich不等式在数理统计中有广泛的应用,Marshall和Olkin把这个不等式推广到矩阵形式,本节将其推广到了一般形式,扩大了它的适用范围。其次,在本章第三节介绍了约束条件下矩阵迹不等式及其统计应用。矩阵特征值是矩阵论中一个.重要内容,它在许多方面都有广泛的应用。本节主要讨论一类特殊形式的矩阵特征值和它的不等式,把已有结果推广到了一般形式。这些结果在数理统计中是十分有用的。3.第五章简单介绍了Moore-Penrose广义逆在多元分析中的应用。多元分析的一个重要内容就是研究随机向量之间的关系,本章主要探讨了随机向量的典型相关系数和广义相关系数之间的关系,给出了随机向量之间典型相关系数和广义相关系数的一些结果

【英文摘要】 Matrix theory has come into wide use in many branches of the modern statistics and has become an indispensable tool in the statistics, moreover their developments have been caused by many concerning problems proposed in the statistics. In this paper, some further problems related statistics are introduced in the matrix theory. Besides, some applications of the above results are also discussed in the statistics, especially in the parameter estimate of the linear model and in the multivariate. The results are extension forms on the basis of the original theory.The main results of this paper are listed in the following.1. In chapter three, we discuss the partial orderings of matrix and its application in the comparison of the model. Firstly, we simply introduce the basic knowledge of the comparison of estimate and model. Then, we use matrix partial order theory to compare the generalized ridge estimation with the least square estimation in section III of this chapter. In some literature, .the problem for the superiority of ridge estimation over least square estimation is studied in the sense of mean square error; in this section, we use the Lowner partial ordering to discuss the superiority of the generalized ridge estimation over least square estimation, and the known conclusions have been extended.2. In chapter four, we mainly introduce the two types of matrix inequalities and their applications in statistics. Firstly, in section I of this chapter,we introduce the extensions of the matrix Kantorovich-type inequalities and its application in statistics. Kantorovich inequalities play an important role in mathematical statistics. Marshall and Olkin generalized it to matrix versions. In this section, we give new extensions of the matrix Kantorovich-type inequalities. Its applying scope is expanded. Secondly, in section II of this chapter, we introduce the inequalities of Hermitian matrix trace with constrained condition and its application in statistics. Matrix eigenvalue is an important conception in matrix theory and it has many applications in other subjects. In this section, the trace of the special form matrix is mainly discussed and the inequalities of Hermitian matrix which are extension forms of the known results are gotten. These inequalities are very useful in mathematical statistics.3. In chapter five, we simply introduce the application of Moore-Penrose inverse in multivariate analysis. It is an important content of the multivariate analysis to study the relationship of random vectors. In this chapter, we get some results about canonical correlation coefficient and generalized correlation coefficient of random vector.

【中文关键词】 广义岭估计; Kantorovich不等式; 特征值; 矩阵的迹; 典型相关系数; 广义相关系数
【英文关键词】 generalized ridge estimation ; kantorovich inequality ; eigenvalue ; trace of matrix ; canonical correlation coefficient ; generalized correlation coefficient
毕业论文目录】
摘要 4-5
ABSTRACT 5-6
符号说明 9-10
1 绪论 10-13
    1.1 研究背景及发展现状 10-11
    1.2 本文主要研究工作 11-13
2 预备知识 13-20
    2.1 Hermite 阵 13-14
    2.2 矩阵分解 14-15
    2.3 广义逆矩阵 15-18
    2.4 偏序 18-20
3 矩阵偏序及其在线性模型比较中的应用 20-28
    3.1 引言 20
    3.2 估计和模型的比较 20-22
    3.3 广义岭估计相对于LS 估计的优良性 22-28
4 两类矩阵不等式及其在线性统计中的应用 28-39
    4.1 引言 28
    4.2 Kantorovich 不等式矩阵形式及其统计应用 28-33
    4.3 约束条件下矩阵迹不等式及其统计应用 33-39
5 Moore-Penrose 广义逆在多元分析中的应用 39-44
    5.1 引言 39
    5.2 随机向量的典型相关系数和广义相关系数 39-44
结论 44-45
参考文献 45-48
致谢 48-49
攻读学位期间发表的学术论文目录 49-50

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