测量误差回归模型下的经验似然估计及其应用

Because people in the data collection process often produce measurement errors, so the project mainly discuss empirical likelihood inference in semiparametric regression model with index and the index variables are measured with errors, partially time-varying coefficient model in which some of the covariates are measured with errors and longitudinal partially linear regression model where the covariates are measured with errors. Meanwhile in our real life, we often encounter dependent data, especially for sequentially collected economic data, which often exhibit evident dependence. So we assume that the data exhibit some kind of dependency structure when we study the above models. While deriving asymptotic properties of unknown parameters, the confidence regions (intervals) of unknown parameters can be obtained. Finally, we will open up the above models in applications to economic, finance, biology and so on. To study the above models by empirical likelihood method, it not only enlarges its application areas, but also enriches theoretical research contents of mathematical statistics.

由于人们在收集数据过程中经常会有所谓的测量误差(Measurement Error)产生，因此本课题主要研究含指标项半参数回归模型中指标变量具有测量误差、时变系数部分线性回归模型中协变量具有测量误差以及纵向数据部分线性回归模型中协变量具有测量误差时的经验似然推断问题。同时在我们的现实生活中，经常遇到相依数据，特别是连续收集的经济数据存在明显的相依结构，所以我们在研究上述模型时假设数据具有某种相依结构。在导出未知参数渐近性质的基础上，进而获得未知参数的置信域(区间)。最后我们开辟上述模型在其他领域的应用，诸如经济、金融、生物等。利用经验似然方法研究此类模型，不仅可以扩大其应用领域，而且也丰富了数理统计的理论研究内容。

本项目主要做了以下工作：.首先我们利用经验似然方法研究了半参数变系数异方差部分线性测量误差回归模型中未知参数的置信域问题。当误差方差函数为已知和未知情形下，我们分别构造了其经验对数似然比函数。对于每一种情形，我们均获得了非参数版本的Wilks定理。这一结果可被用来构造参数的置信域。模拟验证了我们估计方法的优越性。..其次，我们研究了$\alpha$-混合误差下纵向数据回归模型的未知参数的大样本性质。引入了经验对数似然比函数并证明其极限分布为一列加权的中心卡方分布。同时，我们获得了调整的经验对数似然比函数证明了其渐近分布为中心卡方分布。通过大量的计算机计算，表明我们的方法有很好的覆盖比例和区间长度, 同时完全克服了正态置信区间的对称性缺点。