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.

中央限制定理（CLT）是概率理论的基本结果，断言，小型和近似独立的随机数量的大集合的总体行为遵循了普遍的定律，非正式地称为“钟形曲线”。 CLT近似是统计推断的基石，因为它提供了绝大多数统计方法的理论基础，用于估计，假设检验和置信区间。它也通常用于整个科学和许多工业应用中的不确定性评估。尽管广受欢迎，但大多数现有的CLT结果仍无法在数学上完全表达现代大型数据集的典型复杂程度。此外，它们不足以进行高维统计建模。结果，在高维设置中，依赖经典的CLT近似值以验证统计程序的有效性不再是合理的。相反，新的，更精致的CLT保证是有序的。最近的突破性进步导致了新的高维CLT（HDCLT）的制定，后者的有效性具有相当限制的假设。该项目的广泛目标是开发新的HDCLT近似值，这些近似值在更广泛的条件和环境中适用，并阐明其在涉及大型和复杂数据的高维统计问题中的用途。该项目的研究组成部分包括五个主要研究目的：（1）在弱小的条件下生成较弱的观测值，以使其在弱小的条件下产生允许重型数据和单曲数据的独立观察，以实现HDCLTS。 （2）推导新的高维edgeworth扩展； （3）获得新的HDCLT，以进行依赖的随机向量和时间序列过程的总和； （4）研究高维随机矩阵的HDCLT，例如样品协方差矩阵及其光谱； （5）在针对FDR和FWER控制的多次测试问题中部署HDCLT。该项目的研究成果将以重要的方式提高HDCLT近似的理论和应用，并将导致新颖的，可行的工具，用于推断高维统计数据。该奖项反映了NSF的法定任务，并被认为值得通过基金会的知识分子和更广泛的影响来审查审查标准来通过评估来获得支持。