Mathematical Sciences: Stochastic Inequalities, Dependence Structures, and Some Associated Limited Theorems

9504614里诺特摘要 建议的研究中心随机不等式和依赖结构，出现在几个方面：a）多个测试的正相关的测试统计量，以取代已知的错误概率率的独立测试的有用的界限，适当的依赖测试的目标。B）非独立随机变量的预言不等式，它本质上提供了对连续观测数据中未来信息价值的度量。c）正相关不等式和负相关不等式：证明了由随机图上的某些分布得到的随机变量的相关不等式和负相关不等式与有趣的行列式不等式等价。特殊情况已得到直接证明，并通过数值计算。另一个猜想涉及积分不等式，它扩展了著名的货车登伯格-凯斯滕猜想（传闻最近已被证明），并提出了有趣的新应用。新aonjecture的特殊情况下已经建立了使用FKG型和重排不等式，数值计算似乎支持这些扩展。d）相依随机变量的极限定理，目的是使用Stein方法和经典方法，根据相依结构的有意义特征获得收敛速度。 抽样测量的统计独立性假设是许多经典数据分析的基础。当人们可以假设数据来自重复的受控实验，其中重复之间没有误差的遗留时，这是相关的。然而，具有更复杂数据的情况是普遍存在的。拟议的研究中心的某些方面的样本，其中包括观察（随机变量），这是不独立的，和各种模型的依赖结构的研究。这种结构也与物理学中的各种模型有关， 渗透理论，例如，也进行了研究。这项研究的另一个方面与统计大样本理论有关。统计学理论的一个重要部分是基于近似，而近似只对大样本有效。研究了非独立数据的近似方法，以确定该理论在复杂情况下的有效性。近似值的质量在决定大样本近似值有效所需的样本量时非常重要。

9504614 Rinott Abstract The proposed research centers on stochastic inequalities and dependence structures which arise in several contexts: a) Multiple tests for positively dependent test statistics, with the goal of replacing known error probability rates for independent tests by useful bounds for suitably dependent tests. b) Prophet inequalities for dependent random variables, which essentially provide measures on the value of future information in sequentially observed data. c) Positive and negative dependence inequalities: some conjectures on correlation inequalities and negative dependence inequalities for random variables obtained from certain distributions on random graphs are shown to be equivalent to interesting determinental inequalities. Special cases have been proved directly, and by numerical calculations. Another conjecture concerns integral inequalities which extend the well-known van Den Berg-Kesten Conjecture (rumored to have been proved recently) and suggest interesting new applications. Special cases of the new aonjecture have already been established using FKG type and rearrangement inequalities; numerical calculations seem to support these extensions. d) Limit theorems for dependent random variables, with the goal of obtaining convergence rates in terms of meaningful characteristics of the dependence structures, using Stein's method and classical approaches. The assumption of statistical independence of sampled measurements is basic to much of classical data analysis. It is relevant when one can assume that the data arises from repeated controlled experiments in which there is no carryover of errors between repetitions. However, situations with more complex data are ubiquitous. The proposed research centers on the study of certain aspects of samples which consist of observations (random variables) which are not independent, and various models for dependence structures. Such structures are also relevant in various models in physics, and certain issues arising in percolation theory, for example, are also studied. Another aspect of this research is relevant to statistical large sample theory. A good portion of statistical theory is based on approximations which are valid only for large samples. Methods for such approximations for dependent data are investigated, in order to determine the validity of the theory in complex situation. The quality of the approximations is important in deciding how large samples must be in order for large sample approximations to be valid.