Mathematical Sciences: Regression Quantile Methods and Asymptotic Statistical Theory

研究目标包括扩展回归分位数方法和开发新的渐近方法来处理大参数维度的情况。方法论目标包括：(A)在线性回归模型中开发在关于误差分布的任何假设下都是完全有效的自适应估计器，(B)定义和调查离群值检测过程，并将其应用于抵抗离群值和有影响的观测的存在，(C)研究回归分位数算法的性能，以及(D)对计量经济学中的非线性回归模型和结构模型进行稳健分析。新的渐近方法的目标包括：(A)在线性模型中获得回归分位数方法的适当的渐近结果，(B)研究用于分析结构模型的标准和稳健的方法，以及(C)研究这种渐近近似在真实和模拟数据集上的充分性。统计领域的这项研究是为了开发和分析稳健的计量经济学程序。稳健的统计方法是即使应用于数学中的基本假设不切实际和不成立的情况下也能很好地发挥作用的程序。由于经济的数学模型是非常复杂的系统的近似，其真实的基础数学表示很可能是未知的，而且由于数据本身的创建会带来测量误差和其他不确定性，因此拥有已知在不利条件下表现良好的统计技术是有利的。

The research objectives involve extending the regression quantile methodology and developing new asymptotic approaches to cases of large parametric dimension. Methodological aims include the following: (a) to develop adaptive estimators in the linear regression model which are fully efficient under any assumption on the error distribution, (b) to define and investigate outlier detection prodedures and to apply them for providing resistance to the presence of outliers and influential observations, (c) to investigate the performance of the regression quantile algorithm, and (d) to obtain robust analyses for nonlinear regression models and for structural models in econometrics. Aims for new asymptotic approaches include the following: (a) to obtain appropriate asymptotic results for the regression quantile methodology in linear models, (b) to study standard and robust methods for analyzing structural models, and (c) to investigate the adequancy of such asymptotic approximations for real and simulated data sets. This research in the field of statistics is to develop and analyze robust econometric procedures. Robust statistical methods are procedures that work well even when applied to situations where the underlying assumptions in the mathematics are unrealistic and do not hold. Since mathematical models of the economy are approximations of very complex systems for which the true underlying mathematical representation may well be unknown, and since data brings measurement error and other uncertainty with its very creation, it is advantageous to have statistical techniques that are known to perform well under adverse conditions.