线性模型中岭估计和M估计的性质及应用研究
发布时间：2018-08-11 19:16
【摘要】：线性模型被广泛的应用在物理学、经济学、生物学、遗传学、博弈论等领域,其含参估计模型及其相应的有偏估计方法是使用数学求解实际问题这一领域研究的经典课题之一。本文针对有偏估量中的岭参数估计和M—估计两类参数估量方式进行探讨,并得到少许新的论断。在对于有偏估计中的岭估计章节中,对可以进行估量的线性函数的最小二乘估计(LSE)进行了介绍,并概况归纳了该方法的优缺点。在这些基础上,给出在满足Gauss-Markov假设的模型下有偏估计岭估计的概念及性质;探讨了利用变换参数形式对岭估计进行改进的方法;由于复共线性的存在,使用最小二乘估计方法经常会造成估计参数在精度上的损失,引进相对效率的概念,以线性回归模型为例,基于最小特征值和均方误差,定义了两种新的相对效率,讨论了它们的上下界,并证明了在新定义的效率下,岭估计的效率高于最小二乘估计的效率。在关于有偏估计中的M—估计的参数估计章节中,介绍了M—估计方法的相关性质以及其国内外研究现状。在一个新的估计类中,结合岭估计思想,改进了M—估计方法,探讨了改进估计与M—估计的关系,同时,基于5个假设,探究了其是否具有相合性以及收敛性,文章最后证明结合估计具有相合性与收敛性。
[Abstract]:Linear model is widely used in physics, economics, biology, genetics, game theory and so on. Its parametric estimation model and its corresponding biased estimation methods are one of the classical research topics in the field of solving practical problems with mathematics. In this paper, the methods of ridge parameter estimation and M- estimation in biased estimation are discussed, and some new conclusions are obtained. In the chapter of Ridge estimation in biased estimation, the least square estimation of linear function is introduced, and the advantages and disadvantages of this method are summarized. On the basis of these, the concept and properties of biased ridge estimation under the model satisfying Gauss-Markov hypothesis are given, and the method of improving ridge estimation by means of transformation parameter form is discussed, because of the existence of complex collinearity, Using the least square estimation method often results in the loss of the precision of the estimation parameters. The concept of relative efficiency is introduced. Taking the linear regression model as an example, two new relative efficiencies are defined based on the minimum eigenvalue and mean square error. In this paper, the upper and lower bounds are discussed, and it is proved that the efficiency of ridge estimation is higher than that of least square estimator under the new definition. In the chapter on parameter estimation of M- estimation in biased estimation, the related properties of M- estimation and its research status at home and abroad are introduced. In a new class of estimators, we improve the M- estimation method and discuss the relationship between the improved estimators and the M- estimators. At the same time, based on five assumptions, we discuss whether the M- estimators are consistent and convergent. Finally, it is proved that the combined estimation is consistent and convergent.
【学位授予单位】：青岛科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
本文编号：2177981
[Abstract]:Linear model is widely used in physics, economics, biology, genetics, game theory and so on. Its parametric estimation model and its corresponding biased estimation methods are one of the classical research topics in the field of solving practical problems with mathematics. In this paper, the methods of ridge parameter estimation and M- estimation in biased estimation are discussed, and some new conclusions are obtained. In the chapter of Ridge estimation in biased estimation, the least square estimation of linear function is introduced, and the advantages and disadvantages of this method are summarized. On the basis of these, the concept and properties of biased ridge estimation under the model satisfying Gauss-Markov hypothesis are given, and the method of improving ridge estimation by means of transformation parameter form is discussed, because of the existence of complex collinearity, Using the least square estimation method often results in the loss of the precision of the estimation parameters. The concept of relative efficiency is introduced. Taking the linear regression model as an example, two new relative efficiencies are defined based on the minimum eigenvalue and mean square error. In this paper, the upper and lower bounds are discussed, and it is proved that the efficiency of ridge estimation is higher than that of least square estimator under the new definition. In the chapter on parameter estimation of M- estimation in biased estimation, the related properties of M- estimation and its research status at home and abroad are introduced. In a new class of estimators, we improve the M- estimation method and discuss the relationship between the improved estimators and the M- estimators. At the same time, based on five assumptions, we discuss whether the M- estimators are consistent and convergent. Finally, it is proved that the combined estimation is consistent and convergent.
【学位授予单位】：青岛科技大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O212
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