Optimal Shrinkage Estimation for Heteroscedastic Data

收缩估计器是一种强大的统计方法，在科学和工程应用中有着深远的影响。它们提供了有效的工具，将相关群体的信息汇集在一起进行同时推理收缩估计器的应用实例包括医疗保健系统的分析(医院服务质量等)、教育计划的分析(教学计划的有效性)、医疗评估(汇集来自多项研究或临床试验的信息)、多个基因与疾病的关联的排名以及多个制造工艺的比较。研究异方差数据下的收缩估计问题，旨在寻找在不同环境下进行收缩估计的最优方法，研究和识别异方差数据在参数和半参数设置下的风险最优收缩估计。除了深入的理论研究外，该项目还包括全面的数值研究和实际应用。研究主题包括非高斯指数族异方差数据的最优收缩估计量的研究；位置尺度族异方差数据的最优收缩估计量的研究；以及线性回归模型中异方差数据的最优收缩估计的研究。本研究旨在促进对收缩估计量的理论理解，并为医学、自然科学、社会科学和工程学中的数据分析提供强大的技术。

Shrinkage estimators are powerful statistical methods that have profound impact in scientific and engineering applications. They provide efficient tools to pool information from related populations for simultaneous inference -- the data on each population alone often do not lead to the most effective estimation, but by pooling information from the related populations together one can often obtain more accurate estimates for each individual population. Examples of the applications of shrinkage estimators include the analysis of healthcare systems (quality of hospital services, etc.), the analysis of education programs (the effectiveness of teaching programs), the assessment of medical treatments (pooling information from multiple studies or clinical trials), the ranking of multiple genes on their association with diseases, and the comparison of multiple manufacturing processes. This research project will investigate shrinkage estimations in the context of heterogeneous data, aiming to identify optimal ways to conduct shrinkage estimation in various settings.This research project studies and identifies risk-optimal shrinkage estimators under parametric as well as semi-parametric settings for heteroscedastic data. In addition to thorough theoretical investigation, the project includes comprehensive numerical studies and real applications. The research topics include the investigation of optimal shrinkage estimators for heteroscedastic data from non-Gaussian exponential families; the investigation of optimal shrinkage estimators for heteroscedastic data from location-scale families; and the investigation of optimal shrinkage estimators for heteroscedastic data from linear regression models. The research aims to advance the theoretical understanding of shrinkage estimators and also provide powerful techniques for the analysis of data in medical, natural and social sciences, and engineering.