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

9504614 Rinott摘要所提出的研究集中在以下几种情况下的随机不等式和相依性结构：a)正相依检验统计量的多重检验，其目的是用适当相依检验的有用界来取代独立检验的已知错误概率。B)相依随机变量的先知不等式，它实质上提供了对连续观测数据中未来信息的价值的度量。C)正、负相关不等式：由随机图上的某些分布得到的关于随机变量的相关不等式和负相关不等式的一些猜想与有趣的行列式不等式等价。特例已被直接证明，并通过数值计算得到证明。另一个猜想涉及积分不等式，它推广了著名的van Den Berg-Kesten猜想(传闻最近已被证明)，并提出了有趣的新应用。利用FKG型和重排不等式已经建立了新猜想的特例；数值计算似乎支持这些推广。D)相依随机变量的极限定理，目标是利用Stein方法和经典方法，根据相依结构的有意义的特征来获得收敛速度。抽样测量的统计独立性假设是许多经典数据分析的基础。当一个人可以假设数据来自重复的对照实验，其中在重复之间没有误差传递时，这一点是相关的。然而，具有更复杂数据的情况随处可见。建议的研究集中于研究样本的某些方面，包括不独立的观测(随机变量)和各种相依结构模型。这种结构在物理学的各种模型中也是相关的，例如，渗流理论中出现的某些问题也被研究过。这项研究的另一个方面与统计大样本理论有关。统计理论的很大一部分是基于仅对大样本有效的近似。为了确定理论在复杂情况下的有效性，对相依数据的近似方法进行了研究。为了使大样本近似有效，近似的质量在决定大样本必须有多大时很重要。

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.