Mathematical Sciences: Problems in Statistical Modeling

该项目将涉及五个统计建模领域的工作：(1)基于局部多项式回归的非参数估计的高效带宽选择；(2)基于正交阵列抽样的风险评估；(3)基于工程质量改进和生产率的方差建模；(4)基于变量误差的稳健回归；(5)基于动态规划的自适应样条近似。首席研究员和他的研究生将研究统计学在工程和环境科学中的几个应用。第一个应用是对两个量之间的关系进行建模，例如，合金中的碳百分比和硬度，当这种关系没有已知的数学形式时，必须从数据分析中推导出数学形式。第二是评估环境政策的风险（比如癌症），而许多相关的数量（比如癌症和二恶英之间的剂量反应）都是不确定的。第三是工程实验的设计，学习如何减少制造过程中的变化。第四个是在统计建模领域，当一些数据受到严重测量误差的影响时——目标是识别严重误差，或者至少消除它们对感兴趣的数量估计的影响。第五是一种新的计算方法，用于寻找控制系统的最佳方法，例如，需求不确定的企业的“最佳”库存策略。在这里，“最佳”政策恰当地平衡了过剩库存的成本和需求超过库存时的销售损失。

The project will involve work in five areas of statistical modeling: (1) efficient bandwidth selectors for nonparametric estimation by local polynomial regression, (2) risk assessment by orthogonal array sampling, (3) variance modeling for quality improvement and productivity in engineering, (4) robust regression with errors-in-variables, and (5) adaptive spline approximations in dynamic programming. The principle investigator and his graduate students will investigate several applications of statistics to engineering and environmental science. The first application is to modeling the relationship between two quantities, for example, percent carbon in an alloy and hardness, when there is no known mathematical form for this relationship so that the mathematical form must be derived from data analysis. The second is the assessment of risks (say of cancer) from environmental policies when many of the relevant quantities (say the dose response between cancer and dioxin) are known only with uncertainty. The third is the design of engineering experiments to learn how to reduce variation in manufacturing processes. The fourth is in the area statistical modeling when some of the data are subject to gross measurement errors---the object is to identify the gross errors or at least to eliminate their effect upon the estimation of quantities of interest. The fifth is a new computational method for finding the best method of controlling systems, for example, the ``best'' inventory policy for a business where demand is uncertain. Here the ``best'' policy properly balances the cost of excess inventory with the loss of sales when demand exceeds inventory.