测量误差数据下约束线性模型的有偏估计及变量选择研究

Linear model is a widely used parameter model in data analysis. Since in practice, regression model always exists some restrictions and measurement error, which study the restricted linear model with measurement error is practical significance. In this project we mainly discuss the parameter estimation and variable selection of the restricted linear model with measurement error. Firstly, we study the biased estimator of the linear model with exact linear restrictions, when we suspect the linear restrictions is satisfied or not, study the large sample preliminary test estimator based on the Wald test, Likelihood Ratio (LR) test and Lagrangian Multiplier (LM) test, and we discuss the asymptotic normality, consistency of these estimators; secondly, we study how to use the Boosting method to achieve variable selection, and mainly discuss the selection of the biased parameter in the restricted estimators. Through the work of this project, not only enrich the theory of linear model, but also can promote the use of the linear model in the fields of econometrics and medicine.

线性模型是数据分析中应用广泛的一类参数模型。由于实际应用中回归模型经常存在测量误差和一些约束条件，因此研究具有测量误差数据的约束线性模型更具实际意义。本项目主要研究带测量误差数据下约束线性模型的参数估计和变量选择理论与方法。第一，研究模型在等式约束下的参数有偏估计，当约束条件值得怀疑的时候，研究未知参数基于Wald检验、似然比(LR)检验和拉格朗日(LM)检验等大样本检验的预检验估计，并研究估计的相合性和渐近正态性；第二，研究如何利用Boosting方法来实现估计的变量选择，并且重点研究约束有偏估计中偏参数的选取问题。通过本项目的工作，不仅能丰富线性模型的理论，且能促进线性模型在计量经济和医学等领域的广泛应用。

本项目主要研究了测量误差数据下约束线性模型中未知参数的有偏估计理论。首先，针对线性模型中存在的复共线性问题，讨论了模型在等式约束或随机约束下参数的估计问题和估计的效率问题，重点研究约束几乎无偏Liu型估计和两种相对效率；其次研究了线性部分带有测量误差的线性模型，利用几乎无偏估计的思想，讨论了几乎无偏Liu估计；同时对约束条件不确定时，进一步研究参数基于 Wald 检验预检验估计，并重点讨论其偏差和均方误差的比较。