Sharp Inequalities for Sums and Functions of Dependent Variables

在这个项目中，首席调查员(PI)介绍了在研究独立和相依随机变量的和和函数的概率和统计性质时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)建立尖锐的不等式。此外，PI打算进一步开发一种新的方法来近似过程(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被用作开发非参数估计量的关键数量，以及它们被用作创建可信区间和检验假设的关键数量。例如，t统计量是一种自归一化的无单位估计器，通常用于检验关于具有未知方差的分布的平均值的假设。对自归一化估计量的精确结果感兴趣的部分原因是，在需要放松对变量的假设(例如，独立性、正态分布和/或相同分布)的情况下，需要近似p值和检验的能力。对多线性形式和与U-统计量有关的结果的研究涉及到它们在某些随机积分的发展中作为基石的使用，以及它们作为统计学中典型的无偏估计的使用。此外，独立随机变量中的多线性形式的和经常被用来逼近移动平均的非线性估计，这在计量经济学研究中具有重要意义。关于开发新的和改进的统计估计器研究工具(在宽松的假设下)的拟议工作，对于评估在医学和社会科学以及工程学中具有直接影响的假设非常重要，因为它与在比目前可能的情况更广泛的情况下比较相互竞争的治疗方法和技术有关。在历史证据的基础上，对随机过程到达边界需要多长时间的研究，在自然科学和经济学中具有潜在的重要意义，包括研究1)龙卷风袭击城市，2)一个人患癌症，3)地震发生或4)股市崩盘

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