线性模型中Bayes线性无偏最小方差估计及其小样本性质
发布时间：2018-11-08 18:13
【摘要】：在统计学中,线性模型是一个简单且被普遍应用的模型,它已经被广泛应用于商业、工业以及经济学等重要领域.在研究线性模型时,我们首先考虑的是参数估计,人们最早提出的是最小二乘估计.然而,随着随机变量个数的增加,最小二乘估计会出现均方误差变大的缺陷,在各研究领域的实际问题就会出现很大偏差.为此,人们提出Bayes线性无偏最小方差估计以及一系列的有偏估计,如Stein估计、James-Stein估计、岭估计、Liu估计等.本文研究的是线性模型中的参数估计问题,很多学者都对Bayes线性无偏最小方差估计的优良性质进行了研究,同时也与最小二乘估计、广义最小二乘估计、岭估计等进行了比较.本文则在前人的研究基础之上,讨论Bayes线性无偏最小方差估计与Liu估计和James-Stein估计的关系,主要分为以下几部分:首先在绪论中介绍线性模型的基础知识及几种常见估计的发展进程,其中详细介绍了 Bayes线性无偏最小方差估计.第二章主要讨论了 Bayes线性无偏最小方差估计在广义均方误差准则下的性质,将其与Liu估计和James-Stein估计进行比较.第三章将在平衡损失风险函数下,继续讨论Bayes线性无偏最小方差估计的性质,同样将其与Liu估计和James-Stein估计的风险函数进行比较.
[Abstract]:In statistics, linear model is a simple and widely used model, it has been widely used in business, industry, economics and other important fields. When we study the linear model, we first consider the parameter estimation, and the least square estimation is the earliest one. However, with the increase of the number of random variables, the least square estimation will have the defect of increasing the mean square error. Therefore, Bayes linear unbiased minimum variance estimators and a series of biased estimators, such as Stein estimators, James-Stein estimators, ridge estimators, Liu estimators, are proposed. In this paper, the parameter estimation problem in linear model is studied. Many scholars have studied the excellent properties of Bayes linear unbiased minimum variance estimation, and compared it with the least square estimation, the generalized least square estimate and the ridge estimate. In this paper, on the basis of previous studies, we discuss the relationship between Bayes linear unbiased minimum variance estimator and Liu estimator and James-Stein estimator. It is mainly divided into the following parts: firstly, the basic knowledge of linear model and the development process of several common estimators are introduced in the introduction, in which the Bayes linear unbiased minimum variance estimation is introduced in detail. In chapter 2, we discuss the properties of Bayes linear unbiased least variance estimator under the generalized mean square error criterion, and compare it with Liu estimator and James-Stein estimator. In chapter 3, we continue to discuss the properties of Bayes linear unbiased minimum variance estimator under the balanced loss risk function, and compare it with the risk function of Liu estimate and James-Stein estimate.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
本文编号：2319277
[Abstract]:In statistics, linear model is a simple and widely used model, it has been widely used in business, industry, economics and other important fields. When we study the linear model, we first consider the parameter estimation, and the least square estimation is the earliest one. However, with the increase of the number of random variables, the least square estimation will have the defect of increasing the mean square error. Therefore, Bayes linear unbiased minimum variance estimators and a series of biased estimators, such as Stein estimators, James-Stein estimators, ridge estimators, Liu estimators, are proposed. In this paper, the parameter estimation problem in linear model is studied. Many scholars have studied the excellent properties of Bayes linear unbiased minimum variance estimation, and compared it with the least square estimation, the generalized least square estimate and the ridge estimate. In this paper, on the basis of previous studies, we discuss the relationship between Bayes linear unbiased minimum variance estimator and Liu estimator and James-Stein estimator. It is mainly divided into the following parts: firstly, the basic knowledge of linear model and the development process of several common estimators are introduced in the introduction, in which the Bayes linear unbiased minimum variance estimation is introduced in detail. In chapter 2, we discuss the properties of Bayes linear unbiased least variance estimator under the generalized mean square error criterion, and compare it with Liu estimator and James-Stein estimator. In chapter 3, we continue to discuss the properties of Bayes linear unbiased minimum variance estimator under the balanced loss risk function, and compare it with the risk function of Liu estimate and James-Stein estimate.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
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