Collaborative Research: Renyi Divergence-based Robust Inference in Regression, Time Series and Association Studies.

这一合作研究项目的重点是开发一种新的方法，在回归、时间序列和多变量关联研究中基于Rényi分歧家族进行降维，其中心主题是提供对数据污染具有内在健壮性的估计器，仅保持最小的效率损失。这个族不仅刻画了回归和时间序列中充分降维概念背后的条件独立性，而且刻画了多元关联研究中典型变量之间的独立性。该方法的新颖之处在于利用了一个族的调整参数，该参数平衡了估值器的效率和稳健性程度。在这三个领域中的每一个领域，本项目都侧重于调查一系列问题，例如：(I)估计的计算，(Ii)真实维度的检测，(Iii)最佳调谐参数的选择，以及(Iv)通过理论对该方法的形式证明。此外，该项目还通过影响函数和样本/经验影响函数对稳健性进行了深入研究。最后，该项目的重点是寻找一种既稳健又有效的最优Rényi发散度，而不需要事先检测或去除离群值。技术的快速进步导致了大多数科学中的信息过载。许多当代数据集的一个典型特征是它们本质上是相对高维的。这促使应用科学转向一种不同的关系--回归、时间序列和多变量关联的研究流派，通常被称为降维，其目标是降低变量的维度，作为数据分析的第一阶段。然而，高维数据集中离群值的存在对现有降维方法的性能产生了不利影响，导致得出的结论并不完全可靠。考虑到在高维数据集中经常会遇到离群点，并且很难检测到离群点的存在，迫切需要确定对离群点具有一定程度的自动稳健性或不敏感性的降维方法。拟议的项目提供了稳健的降维方法，这将大大有助于分析社会科学、机器学习、体育、经济学、环境研究、形态计量学和癌症研究等领域产生的高维数据。事实上，该项目不仅将为各学科的科学家提供新的工具，以获得关于高维数据分析的可靠结论，而且将显著推进统计理论，从而为降维铺平一条新的研究道路。

This collaborative research project focuses on developing a novel approach to dimension reduction in regression, time series, and multivariate association studies based on a family of Rényi divergences, with a central theme of providing estimators that are inherently robust to data contamination, sustaining only a minimal loss in efficiency. This family not only characterizes the conditional independence underlying the concept of sufficient dimension reduction in regression and time series, but also characterizes independence between canonical variates in multivariate association studies. The novelty of the approach lies in exploiting a tuning parameter of the family, which balances the efficiency and the degree of robustness of the estimators. In each of the three areas, this project focuses on investigating a host of issues such as: (i) the computation of estimates, (ii) the detection of the true dimension, (iii) the selection of an optimal tuning parameter, and (iv) a formal justification of the method via theory. Furthermore, the project focuses on carrying out an in-depth study of robustness via influence functions and sample/empirical influence functions. Finally, the project focuses on finding an optimal Rényi divergence measure that is both robust and efficient, without the need for prior outlier detection or removal. Rapid advances in technology have led to an information overload in most sciences. A typical characteristic of many contemporary datasets is that they are relatively high-dimensional in nature. This has prompted a shift in the applied sciences toward a different relationship-study genre arising in regression, time series and multivariate association, popularly known as dimension reduction, whose goal is to reduce the dimensionality of the variables as a first phase in the data analysis. However, the presence of outliers in high-dimensional datasets adversely affects the performance of existing dimension reduction methodologies, resulting in conclusions that are not completely reliable. Given that outliers are commonly encountered in high-dimensional datasets and that their presence is hard to detect, there is an urgent need to identify dimension reduction methods that possess some degree of automatic robustness, or non-sensitivity, to outliers. The proposed project provides robust dimension reduction methods, which would contribute significantly to the analysis of high-dimensional data arising in fields such as the social sciences, machine learning, sports, economics, environmental studies, morphometrics and cancer studies, among others. In fact, this project will not only provide novel tools for scientists in various disciplines to obtain reliable conclusions on high-dimensional data analysis, but also significantly advance the statistical theory, thereby paving a new research path in dimension reduction.