导　　师： 刘万荣

学科专业： 070103

授予学位： 硕士

作　　者： ;

机构地区： 湖南师范大学

摘　　要： EV/(errors-in-variables/)模型,是自变量和因变量都带有误差的回归模型。EV模型的研究有很长的一段历史。早在20世纪以前,科学家们就已经关注此模型/(Adcock,1877,1878;Kummel,1879/)。Fuller/(1987/)出版了专著,主要讨论了线性EV模型。但是,由于EV模型的特殊结构,对它的研究要比经典的回归模型困难,EV模型的参数估计的存在性及其相合性问题比经典的回归模型要复杂得多/(Cheng/&Van Nass,1999/)。迄今为止,人们对EV模型的研究主要有以下几种途径。/(一/)对模型作一些假定,常见的是假定测量误差服从正态分布且方差满足某些条件/(如协方差阵为正定矩阵等/),在此假定下研究模型的参数估计及其性质。也有些学者研究了误差为非正态分布的情形。/(二/)为了避免这些人为的假定而采用有重复观测/(replicated observations/),利用这些观测数据得到具有良好性质的相关估计。 变系数模型/(Varying-coefficient Models/)自Cleveland,Grosse,和Shyu/(1992/)与Hastie,Tibshirani/(1993/)首次提出至今,已在国内外产生了较深刻的影响。理论上得到了较深入的研究,实践上也被广泛地应用于生物、医学等方面。 由于在实际问题中自变量的观测存在不可忽视的误差/(如测量工具等因数引起的误差等/)。因此,将变系数模型和EV模型加以结合就更接近实际,且具有很好的理论意义和实际应用意义。这就提出了一个新的研究方向:变系数EV模型。本文讨论如下变系数EV模型:/(?/)/(i=1,2,…,n/).其中: x/_i=/(x/_/(i0/),x/_/(i1/),…,x/_/(ip/)/)~T,β/(t/_i/)=/(β/_0/(t/_i/),β/_1/(t/_i/),…,β/_p/(t/_i/)/)~T, u/_i=/(u/_/(i0/),u/_/(i1/),…,u/_/(ip/)/)~T,X/_i=/(X/_/(i0/),X/_/(i1/),…,X/_/(ip/)/)~T. /(x/_i,y/_i/)是R~/(p+1/)×R~1上的随机变量,在实际中,/(x/_i,y/_i/)的值一般不能精确观测,其观测值为/(X/_i,Y/_i/)。β/_j/(t/)/(j=0,1,…,p/)是有界连 Formally the errors-in-variables /(EV/) models , also called measurement error /(ME/) models, are just the regression models with both dependent and independent variables being subjects to error . The study of EV models has a long history . Scientists encountered this model long before the twentieth century /(Adcock , 1877 , 1878;Kummel, 1879/) . Many works are given by the monograph of Fuller /(1987/) , which concems mainly the linear case . But because the special structure of EV models , the study is more difficult and the problem of existence of consistent estimate of parameters in EV model is more complicated than in the classical regression model /(Cheng /& Van Nass , 1999 /) . So far, the usual approach of the study on EV models are this : Firstly , some assumptions are made , for example , the model assumes a normal error and the variance of the measurement errors content some assumptions /(for example , covariance matrix is positive definite matrix etc . /) There exists a huge literature under such assumptions in the model. The study of non-normal case are made by some scholars too. Another approach, aiming at avoiding such some what artificial assumptions is to take replicated observations and based on these observations , to establish estimates with good asymptotic properties .Varying-coefficient models /(VCM/) was propsed by Clevel-and, Grosse and Shyu /(1992/) , then discussed by Hastie and Tibshirani/(1993/) in detail . Up to now , it has made great influence in our world . In theory , the researchs of VCM has been given most attention recently;Furthermore , extensive applications of VCM into biometrics and medicine has implemented successfully .Considering there extist some cannot slighting errors in the observation of independent variable in practice /( for example errors come from measure tools and so on /) , we added EV models to varying-coefficient models ,which may express our problems more externally and the profounding meanings will be made in prcatice and theory. Then we can find a new rese

关 键 词： 变系数模型 模型 变系数 模型 核估计 最小二乘法 相合性 一致强相合性

领　　域： [理学] [理学]