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，主要研究者（PI）在研究独立和依赖随机变量的总和和功能的概率和统计特性的研究中介绍了三个相关的问题领域。特别是，PI提议在自立过程中以及独立变量和因变量中的多线性形式和U统计数据（无偏见的统计）的总和中产生急剧的不平等现象。此外，PI打算进一步开发一种新的方法来近似于击中给定边界的过程（具有一般依赖性结构）的预期时间（具有一般依赖性结构）。对自称过程的研究的兴趣源于它们用作非参数估计量开发的关键数量，以及它们用作创建置信区间和假设检验的关键量的使用。例如，T统计是一种自我归一化的，无单位的估计量，通常用于测试有关具有未知方差的分布平均值的假设。自称估计器对尖锐结果的兴趣部分基于近似p值的需求以及在需要放松变量的假设时（例如独立性，正态性和/或相同的分布），在情况下的测试功能。与多线性形式和U统计数据总和相关的结果的研究与它们用作某些随机积分开发的基础以及它们用作统计数据中典型的无偏估计量有关。此外，独立随机变量中多线性形式的总和经常用于近似移动平均值的非线性估计量，这在计量经济学研究中至关重要。 关于开发新工具（在放松的假设下）进行统计估计器研究的拟议工作对于评估在医学和社会科学中具有直接影响的假设以及通过比较竞争性治疗和技术在更广泛的风景中的比较而进行工程的假设很重要。在历史证据的基础上，关于一个随机过程需要多长时间撞击边界需要多长时间的研究对物理科学和经济学具有潜在的重要含义，包括在研究中需要多长时间造成城市的龙卷风，2）一个人发展癌症，3）地震发生或4）股票市场崩溃或4）崩溃或4）