非参数测量误差模型的统计推断
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摘要
本文主要研究的是在协变量存在测量误差的情形下非参数回归问题的统计推断方法.在许多实际应用领域,统计模型中感兴趣的变量往往由于某些原因不能直接观测,例如：抽样误差,实验误差,工具误差以及调查误差等等.在这种情况下,取而代之的是一个代理变量(相对于真实的协变量而言)被观测到.此时,相比较真实协变量被直接观测到而言,用代理变量来对真实协变量所构建的模型进行基本的统计分析将是非常复杂的.我们把这种协变量存在测量误差的问题称为”测量误差问题”,而分析这些数据的统计模型和方法称之为”测量误差模型”
本文大致可以分为四部分.第一部分简述了关于误差模型和数据结构的一些常用术语以及无差异误差和有差异误差的区别,并简单总结了测量误差回归模型的基本思想和常用的估计方法.
非参数方法现在在各个领域得到了越来越广泛的应用.在测量误差问题研究中,非参数回归问题开始变得极其重要,特别是近二十年来,统计学家们提出了一些估计方法并且已经开始用之解决实际问题.本文的第二部分简单介绍了非参数古典测量误差模型和非参数Berkson测量误差模型的一些重要的估计方法.这些估计方法是非参数测量误差模型中非常经典的方法,并且在实际问题中得到了人们的广泛应用.
本文的第三部分研究了协变量存在Berkson误差或存在混合误差时的非参数回归模型的估计方法.我们通过选择两个简单的密度函数构建一个紧算子,使得回归函数变成了第一类Fredholm积分方程的解.由于这个积分方程的解涉及到病态逆问题,我们基于Tikhonov规则化准则给出了回归函数的一个新估计.我们的估计方法相比较现有的方法在计算上比较简单,对大部分误差分布都适用.
本文的最后一个部分研究了在没有误差模型假设下的非参数测量误差模型的估计问题.此时的观测数据由两个独立的部分组成：一是关于反应变量和代理变量的代理样本,另一个是关于真实变量和代理变量的核实数据.在没有误差模型假设下,我们利用代理样本和核实数据基于正交序列估计和截尾近似方法提出了一个新的非参数回归函数估计.值得一提的是我们的方法可以推广到协变量是多元变量的情形,特别的是部分协变量存在测量误差而其它协变量不存在测量误差的情形.
This paper is about statistical inference for nonparametric regression problems inwhich predictors are measured with error. In many areas of application, statisticallymeaningful models of interest are defined in terms of predictors that for some reason arenot directly observable, for example, sampling error, experimental error, instrumentalerror, error in survey, etc. In such situations, it is not uncommon for surrogate variablesto be observed instead. The surrogate variables for true predictors complicates the fun-damental statistical analysis on which the purpose of the analysis is inference about amodel defined in terms of true predictors. These problems are commonly called mea-surement error problems, and the statistical models and methods used to analyze suchdata are known as measurement error models or errors-in-variables models.
This paper can be divided broadly into four main parts. The first part defines basicterminology of error model, data sources and the distinction between nondiferential anddiferential errors, and gives an overview of the basic ideas and estimation techniques ofregression model with measurement error.
Nonparametric methods are enjoying increased application. The field of nonpara-metric regression has become extremely important in the past twenty years, and in mea-surement error problems techniques are now established. The second part briefly givesan overview of some important techniques in nonparametric regression models when thedata are observed with classical errors or with Berkson errors.
The third part discusses the estimation of nonparametric regression models with theexplanatory variable being measured with Berkson errors or with a mixture of Berksonand classical errors. By constructing a compact operator, the regression function is thesolution of an ill-posed inverse problem, and we propose an estimation procedure basedon Tikhonov regularization.
The fourth part of this paper considers estimation approaches for nonparametricregression measurement error models when both independent validation data on covari- ables and primary data on the response variable and surrogate covariables are available.An estimator which integrates orthogonal series estimation and truncated series approx-imation method is derived without any error model structure assumption between thetrue covariables and the surrogate variables. Most importantly, our proposed methodol-ogy can be readily extended to the case that only some of covariates are measured witherrors with the assistance of validation data.
本文大致可以分为四部分.第一部分简述了关于误差模型和数据结构的一些常用术语以及无差异误差和有差异误差的区别,并简单总结了测量误差回归模型的基本思想和常用的估计方法.
非参数方法现在在各个领域得到了越来越广泛的应用.在测量误差问题研究中,非参数回归问题开始变得极其重要,特别是近二十年来,统计学家们提出了一些估计方法并且已经开始用之解决实际问题.本文的第二部分简单介绍了非参数古典测量误差模型和非参数Berkson测量误差模型的一些重要的估计方法.这些估计方法是非参数测量误差模型中非常经典的方法,并且在实际问题中得到了人们的广泛应用.
本文的第三部分研究了协变量存在Berkson误差或存在混合误差时的非参数回归模型的估计方法.我们通过选择两个简单的密度函数构建一个紧算子,使得回归函数变成了第一类Fredholm积分方程的解.由于这个积分方程的解涉及到病态逆问题,我们基于Tikhonov规则化准则给出了回归函数的一个新估计.我们的估计方法相比较现有的方法在计算上比较简单,对大部分误差分布都适用.
本文的最后一个部分研究了在没有误差模型假设下的非参数测量误差模型的估计问题.此时的观测数据由两个独立的部分组成：一是关于反应变量和代理变量的代理样本,另一个是关于真实变量和代理变量的核实数据.在没有误差模型假设下,我们利用代理样本和核实数据基于正交序列估计和截尾近似方法提出了一个新的非参数回归函数估计.值得一提的是我们的方法可以推广到协变量是多元变量的情形,特别的是部分协变量存在测量误差而其它协变量不存在测量误差的情形.
This paper is about statistical inference for nonparametric regression problems inwhich predictors are measured with error. In many areas of application, statisticallymeaningful models of interest are defined in terms of predictors that for some reason arenot directly observable, for example, sampling error, experimental error, instrumentalerror, error in survey, etc. In such situations, it is not uncommon for surrogate variablesto be observed instead. The surrogate variables for true predictors complicates the fun-damental statistical analysis on which the purpose of the analysis is inference about amodel defined in terms of true predictors. These problems are commonly called mea-surement error problems, and the statistical models and methods used to analyze suchdata are known as measurement error models or errors-in-variables models.
This paper can be divided broadly into four main parts. The first part defines basicterminology of error model, data sources and the distinction between nondiferential anddiferential errors, and gives an overview of the basic ideas and estimation techniques ofregression model with measurement error.
Nonparametric methods are enjoying increased application. The field of nonpara-metric regression has become extremely important in the past twenty years, and in mea-surement error problems techniques are now established. The second part briefly givesan overview of some important techniques in nonparametric regression models when thedata are observed with classical errors or with Berkson errors.
The third part discusses the estimation of nonparametric regression models with theexplanatory variable being measured with Berkson errors or with a mixture of Berksonand classical errors. By constructing a compact operator, the regression function is thesolution of an ill-posed inverse problem, and we propose an estimation procedure basedon Tikhonov regularization.
The fourth part of this paper considers estimation approaches for nonparametricregression measurement error models when both independent validation data on covari- ables and primary data on the response variable and surrogate covariables are available.An estimator which integrates orthogonal series estimation and truncated series approx-imation method is derived without any error model structure assumption between thetrue covariables and the surrogate variables. Most importantly, our proposed methodol-ogy can be readily extended to the case that only some of covariates are measured witherrors with the assistance of validation data.
引文
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