Sharp Inequalities for Sums and Functions of Dependent Variables

在这个项目中，首席研究员（PI）介绍了在研究独立和相关随机变量的和和函数的概率和统计性质中至关重要的三个相关问题领域。特别是，PI提出了自标准化过程的矩的尖锐不等式，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）的和。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在非参数估计器的发展中作为关键量的使用，以及它们作为创建置信区间和假设检验的关键量的使用。例如，t统计量是一种自归一化的无单位估计量，通常用于检验关于方差未知分布的均值的假设。对自归一化估计量的尖锐结果的兴趣部分是基于在需要放松对变量的假设（例如独立性、正态性和/或相同分布）的情况下需要近似p值和检验的能力。研究与多元线性形式和u统计量有关的结果，涉及到它们在发展某些随机积分中的基础，以及它们在统计学中作为典型的无偏估计量的使用。此外，独立随机变量的多线性形式和经常用于逼近非线性移动平均估计量，这在计量经济学研究中具有重要意义。关于（在宽松的假设下）为研究统计估计器开发新的和改进的工具的拟议工作，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更广泛的情况下比较相互竞争的治疗和技术有关。在历史证据的基础上，对随机过程达到边界需要多长时间的研究，在物理科学和经济学中具有潜在的重要意义，包括研究1)龙卷风袭击一座城市需要多长时间，2)一个人患癌症需要多长时间，3)地震发生需要多长时间，4)股市崩溃需要多长时间

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