Central Limit Theorems and Inference in High Dimensions

高维中心极限定理和推理

• 批准号: 2113611
• 负责人: Arun Kuchibhotla
• 金额: $ 30万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2113611&HistoricalAwards=false
• 关键词: Central; Limit; Theorems; Inference; Dimensions

The Central Limit Theorem (CLT) is a fundamental result in probability theory, asserting that the aggregate behavior of large ensembles of small and approximately independent stochastic quantities follows a universal law, informally known as the bell curve. The CLT approximation is a cornerstone of statistical inference, as it provides the theoretical underpinning of the vast majority of statistical methods for estimation, hypothesis testing, and confidence intervals. It is also routinely used for uncertainty assessment across the sciences and in many industrial applications. Despite the immense popularity, most existing CLT results are unable to fully express mathematically the degree of complexity that is typical of modern, large datasets. In addition, they are inadequate for high-dimensional statistical modeling. As a result, reliance on classic CLT approximations to verify the validity of statistical procedures is no longer justifiable in high-dimensional settings. Instead, new, more refined CLT guarantees are in order. Recent breakthrough advances have led to the formulation of new high-dimensional CLTs (HDCLTs), whose validity holds but only under fairly restrictive assumptions. The broad goal of this project is to develop new HDCLT approximations that are applicable across a significantly wider range of conditions and settings and to elucidate their uses in high-dimensional statistical problems involving large and complex data.The research components of this project include five main research aims: (1) to produce HDCLTs for independent observations under weak conditions that allow for heavy-tail data and singular covariances; (2) to derive new high-dimensional Edgeworth expansions; (3) to obtain new HDCLTs for sums of dependent random vectors and time series processes; (4) to study HDCLTs for high-dimensional random matrices, such as the sample covariance matrix, and their spectra; and (5) to deploy HDCLTs in multiple-testing problems targeting FDR and FWER control. The research outcomes of this project will advance in important ways the theory and applications of HDCLT approximations and will lead to novel, practicable tools for inference in high-dimensional statistics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

中心极限定理（英语：Central Limit Theorem，缩写：CLT）是概率论中的一个基本结果，它认为由小且近似独立的随机量组成的大集合的总体行为遵循一个普遍规律，非正式地称为钟形曲线。CLT近似是统计推断的基石，因为它为估计，假设检验和置信区间的绝大多数统计方法提供了理论基础。它也经常用于跨科学和许多工业应用的不确定性评估。尽管非常受欢迎，但大多数现有的CLT结果无法完全表达现代大型数据集的复杂程度。此外，它们不适合高维统计建模。因此，依赖经典CLT近似来验证统计过程的有效性在高维环境中不再合理。相反，新的、更完善的CLT保证是必要的。最近的突破性进展导致制定新的高维CLTs（HDCLTs），其有效性，但只有在相当严格的假设。本项目的主要目标是开发新的HDCLT近似，使其适用于更广泛的条件和设置，并阐明其在涉及大量和复杂数据的高维统计问题中的应用。本项目的研究内容包括五个主要的研究目标：（1）在允许重尾数据和奇异协方差的弱条件下产生独立观测的HDCLT;（2）导出新的高维Edgeworth展开式;（3）获得新的相关随机向量和时间序列过程的HDCLT;（4）研究高维随机矩阵的HDCLT，如样本协方差矩阵及其谱;（5）在针对FDR和FWER控制的多重检验问题中部署HDCLT。该项目的研究成果将以重要的方式推进HDCLT近似的理论和应用，并将为高维统计的推断带来新颖、实用的工具。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。