Sharp Inequalities for Sums and Functions of Dependent Variables

因变量的和与函数的尖锐不等式

• 批准号: 0205791
• 负责人: Victor de la Pena
• 金额: $ 26.13万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0205791&HistoricalAwards=false
• 关键词: Sharp; Inequalities; Sums; Functions; Dependent

0205791de la Pena In this project the Principal Investigator (PI) introduces three related problem areas of key importance in the study of the probabilistic and statistical properties of sums and functions of independent and dependent random variables. In particular, the PI proposes to develop sharp inequalities for the moments of self-normalized processes, as well as for sums of multilinear forms and U-statistics (unbiased statistics) in independent and dependent variables. In addition, the PI intends to further develop a novel approach to approximating the expected time it takes a process (with a general dependence structure) to hit a given boundary. The interest in the study of self-normalized processes stems from their use as key quantities in the development of non-parametric estimators, as well as for their use as pivotal quantities for the creation of confidence intervals and tests of hypothesis. For example, the t-statistic is a self-normalized and unit-less estimator commonly used in the testing of hypotheses about the mean of a distribution with unknown variance. The interest on sharp results for self-normalized estimators is based in part in the need for approximating p-values and the power of tests in situations when the assumptions on the variables need to be relaxed (e.g. independence, normality and/or identical distribution). The study of results related to sums of multilinear forms and U-statistics is related to their use as building blocks in the development of certain stochastic integrals, as well as for their use as the typical unbiased estimators in statistics. Moreover, sums of multilinear forms in independent random variables are frequently used in approximating non-linear estimators of moving averages, which are of fundamental importance in econometric studies. The proposed work concerning the development of new and improved tools (under relaxed assumptions) for the study of statistical estimators is important for the assessment of hypotheses with direct implications in medical and social sciences as well as engineering through its connection to the comparison of competing treatments and technologies under a wider set of scenarios than is currently possible. The study on how long it takes for a random process to hit a boundary, on the basis of historical evidence, has potential important implications in the physical sciences and economics including in the study of how long it will take for 1) a tornado to hit a city, 2) a person to develop cancer, 3) an earthquake to occur or 4) the stock market to crash

0205791de la Pena 在该项目中，首席研究员 (PI) 介绍了在独立随机变量和因随机变量的总和和函数的概率和统计特性研究中至关重要的三个相关问题领域。特别是，PI 建议为自归一化过程的矩以及自变量和因变量中的多线性形式和 U 统计量（无偏统计量）的总和开发尖锐的不等式。此外，PI 打算进一步开发一种新颖的方法来估算流程（具有一般依赖结构）达到给定边界所需的预期时间。对自归一化过程研究的兴趣源于它们在非参数估计量开发中用作关键量，以及用作创建置信区间和假设检验的关键量。例如，t 统计量是一种自归一化且无单位的估计量，通常用于检验有关方差未知的分布均值的假设。自归一化估计量对尖锐结果的兴趣部分是基于在需要放宽变量假设（例如独立性、正态性和/或相同分布）的情况下需要近似 p 值和检验的功效。对与多线性形式和 U 统计量之和相关的结果的研究与它们在某些随机积分的开发中用作构建块以及它们在统计中用作典型无偏估计量有关。此外，独立随机变量的多线性形式之和经常用于近似移动平均线的非线性估计量，这在计量经济学研究中至关重要。 拟议的关于开发统计估计量研究的新的和改进的工具（在宽松的假设下）的工作对于评估对医学和社会科学以及工程学有直接影响的假设很重要，因为它与竞争治疗方法的比较有关以及比目前更广泛的场景下的技术。根据历史证据，研究随机过程到达边界需要多长时间，这对物理科学和经济学具有潜在的重要意义，包括研究 1) 龙卷风到达边界需要多长时间。一座城市，2) 一个人患上癌症，3) 发生地震或 4) 股市崩盘