Exact inequalities and limit theorems for Rademacher and self-normalized sums, and related statistics

Rademacher 和自归一化和的精确不等式和极限定理以及相关统计

• 批准号: 0805946
• 负责人: Iosif Pinelis
• 金额: $ 15万
• 依托单位: Michigan Technological University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805946&HistoricalAwards=false
• 关键词: Exact; inequalities; limit; theorems; Rademacher

The main objectives of the project are as follows: * Prove the longstanding conjecture on the best constant factor in the Rademacher-Gaussian tail comparison. * Prove another longstanding conjecture, on the asymptotic domination of the Rademacher tail by the Gaussian one. * Consider also the ``asymmetric'' case. * Extend to the case of moderate deviations the result due to Shao et al. on the saddle-point approximation to large-deviation probabilities of a self-normalized sum of independent random variables. * Obtain limit theorems, including Berry-Esseen-type bounds and Cramer-type large-deviation asymptotics, for Pearson's product-moment sample correlation coefficient and a number of similar and more general statistics. Thus, the investigator aims to solve longstanding and difficult problems of probability theory and mathematical statistics. The first two of them concern some of the most important properties of such a classical and fundamental object as the Rademacher sums, whose distributions play the role of the extreme points of the set of the distributions of sums (and self-normalized sums) of any independent symmetric random variables. Extensions to the ``asymmetric'' case will also be considered. Closely related are other main objectives of the project, concerning limit theorems for self-normalized sums (or, equivalently, for Student's statistic). The main impact will be in significantly better understanding of important properties of some of the most fundamental objects in probability theory and mathematical statistics. The successful completion of the project will also result in novel and important applications to such classical objects in statistics as Student's test and Pearson's correlation test, which are some of the very few hypotheses tests used most broadly in sciences and engineering. While there are great difficulties to overcome, it appears that the attainment of these objectives is within reach, given a number of advances already made by the investigator and his rather unique expertise in various areas of probability and statistics, as well as his demonstrated abilities to identify and solve difficult and longstanding problems and also to work effectively in a wide and highly diverse range of fields, including mechanical engineering, biology, operations research and combinatorics, and geometry and physics. Efforts will be made to disseminate results, not only via publication in wide-circulation journals, but also via news networks (stories on the investigator's work on evolution modeling and the Eiffel tower shape modeling have already been broadcast around the world by the United Press International and other news agencies). A number of graduate students will be involved into the project; efforts will be made to recruit from underrepresented minorities.

该项目的主要目标如下： * 证明长期以来关于Rademacher-Gaussian 尾部比较中最佳常数因子的猜想。 * 证明另一个长期存在的猜想，即高斯尾巴渐近支配拉德马赫尾巴。 * 还要考虑“不对称”的情况。 * 将 Shao 等人的结果扩展到中等偏差的情况。关于独立随机变量的自归一化和的大偏差概率的鞍点近似。 * 获得极限定理，包括 Berry-Esseen 型界限和 Cramer 型大偏差渐近，用于 Pearson 的乘积矩样本相关系数和许多类似且更一般的统计量。因此，研究者的目标是解决概率论和数理统计中长期存在的难题。其中前两个涉及诸如拉德马赫和这样的经典和基本对象的一些最重要的属性，其分布起着任意的和（以及自归一化和）分布集合的极值点的作用。独立的对称随机变量。还将考虑对“不对称”情况的扩展。密切相关的是该项目的其他主要目标，涉及自归一化和（或等效地，学生统计量）的极限定理。主要影响将是显着更好地理解概率论和数理统计中一些最基本对象的重要属性。该项目的成功完成还将为统计学中的经典对象（如学生检验和皮尔逊相关检验）带来新颖而重要的应用，这些是科学和工程中最广泛使用的极少数假设检验之一。尽管有很大的困难需要克服，但考虑到研究者已经取得的一些进展以及他在概率和统计各个领域相当独特的专业知识，以及他所表现出的能力，这些目标的实现似乎是可以实现的。识别和解决困难和长期存在的问题，并在广泛且高度多样化的领域有效工作，包括机械工程、生物学、运筹学和组合学、几何学和物理学。将努力传播结果，不仅通过在广泛发行的期刊上发表，还通过新闻网络（关于研究者在进化模型和埃菲尔铁塔形状模型方面的工作的故事已经由合众国际社在世界各地播出）和其他新闻机构）。许多研究生将参与该项目；将努力从代表性不足的少数群体中招募人员。