【作者】 刘冬喜；

【导师】 刘万荣；

【作者基本信息】 湖南师范大学 ， 基础数学， 2008， 硕士

【摘要】 线性模型中参数的有偏估计的研究一直是回归分析的热点问题。基于最小二乘法处理病态阵X共线性问题的不足,线性有偏估计是改进最小二乘估计最直接的方法。无约束线性模型中参数的有偏估计理论的发展已经相对成熟,但在大量的统计问题,如试验设计、方差分析模型和协方差分析模型中,由于附加信息等原因,回归参数满足某些约束条件,这使得带线性等式约束的回归分析具有很重要的研究意义和应用价值。经研究发现,现今通用的一般带约束最小二乘估计同最小二乘估计一样,在处理共线性问题上也存在不足。因此近年来,很多学者试图找更好的方法来改进一般带约束的最小二乘估计方法。本文试图寻找带约束线性模型中优于最小二乘估计的约束型有偏估计方法,对约束线性回归模型提出了一种新的参数估计的标准(平均散布误差准则),得到了回归系数一种新的约束型有偏估计——条件部分根方估计(CPRSE),并在缺失数据情形下加以应用,探讨了缺失模型(4.2)回归系数的约束型岭估计(RRE)的性质。第一章综述了目前国内外线性模型参数有偏估计的发展历史和研究现状。第二章给出了一些预备知识。第三章在郭建锋及史建红的基础上作者提出了一种新的可容许估计——条件部分根方估计(CPRSE),证明了存在参数k,可使回归系数β的CPRSE的均方误差(MSE)小于约束最小二乘估计(RLSE)的均方误差;在平均散布误差(MDE)准则下给出了CPRSE优于RLSE的充要条件或充分条件;讨论了确定最优尼值的方法。第四章研究了含缺失数据的约束线性模型,研究了缺失数据的填充法,并进一步对含缺失数据的约束线性模型回归系数建立了条件部分根方估计。第五章给出了缺失模型(4.2)回归系数β的约束型岭估计(RRE),适当地选择参数k,可使回归系数β的约束型岭估计(RRE)的均方误差(MSE)小于约束最小二乘估计(RLSE)的均方误差;在平均散布误差(MDE)准则下给出了RRE优于RLSE的充要条件或充分条件。更多还原

【Abstract】 The research of the biased estimation of parameters in the linear model is all the time one of the most popular issues of regression analysis.While dealing with the multicollinearity of design matrix X,the ordinary least squares estimation is always helpless.The linear biased estimation is the most direct method in ameliorating the ordinary least squares estimation.The development of the biased estimation theory of parameters has been relatively mature in the linear regression model without additional restrictions.In a great deal of statistical problems such as in the experiment design,the models of the variance analysis and the covariance analysis and so on,because of additional information and other reasons,regression parameters meet certain restrictive conditions, which show in practice the investigative significances and the applied values of the restricted linear regression.Like the ordinary least squares estimation,the widely applied ordinary restricted least squares estimation is also disadvantageous for dealing with the multicollinearity of design matrix.As a result,a great many researchers recently try to find out better methods to improve the ordinary restricted least squares estimation.In this dissertation,wetry to seek some biased estimations better than the ordinary restricted least squares estimation in restricted linear regression model.Besides,present a new standard(MDE) for estimating the regression coefficient of restricted linear regression model,and give a new restricted biased estimation of regression coefficient——conditional partial root squares estimation and it is introduced into restricted linear regression model with missing data,and investigate the features of the restricted ridge estimation of regression coefficient in model(4.2) with missing data.In chapter 1,we discuss the history of development and the current situation of the biased estimation in the linear regression model.In chapter 2,we give some pre-knowledge.In chapter 3,on the basis of Guo Jian-Feng and Shi Jian-hong,the author give a new permissible estimation——Conditional partial root squares estimation(CPRSE),prove the existence of parameters k of making mean squares error(MSE) of the Conditional partial root squares estimation less than that of the restricted least squares estimation(RLSE). We gain a necessary and sufficient condition or sufficient conditions which CPRSE is superior to RLSE under the mean dispersion error(MDE) matrix comparisons criterion;and some of methods are discussed to evaluate the optimal value of k.In chapter 4,we research the restricted linear regression model with missing data and imputation methods in missing data,and we have access to conditional partial root squares estimation(CPRSE) of regression coefficient in restricted Linear regression model with missing data.In chapter 5,we give the restricted ridge estimation of regression coefficientβin model(4.2) with missing data and k can be chosen to make mean squares error(MSE) of the restricted ridge estimation(RRE) less than of the restricted least squares estimation(RLSE),we gain a necessary and sufficient condition or sufficient conditions which’ RRE is superior to RLSE under the mean dispersion error(MDE) matrix comparisons criterion.更多还原