Optimal Shrinkage Estimation for Heteroscedastic Data

收缩估计是一种强大的统计方法，在科学和工程应用中具有深远的影响。它们提供了有效的工具来汇集来自相关种群的信息以进行同时推断——单独使用每个种群的数据通常不能得出最有效的估计，但是通过汇集来自相关种群的信息，人们通常可以获得对每个个体种群的更准确的估计。收缩估计器的应用实例包括医疗保健系统的分析（医院服务的质量等），教育计划的分析（教学计划的有效性），医学治疗的评估（从多个研究或临床试验中汇集信息），与疾病相关的多个基因的排名，以及多个制造过程的比较。本研究项目将研究在异构数据背景下的收缩估计，旨在确定在各种设置下进行收缩估计的最佳方法。本研究项目研究并确定在异方差数据的参数和半参数设置下的风险最佳收缩估计器。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。研究课题包括：非高斯指数族异方差数据的最优收缩估计；位置尺度异方差数据最优收缩估计的研究并对异方差数据线性回归模型的最佳收缩估计进行了研究。本研究旨在促进对收缩估计器的理论认识，并为医学、自然科学、社会科学和工程领域的数据分析提供有力的技术。

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.