Probability Limit Theorems and Statistical Applications

概率极限定理和统计应用

• 批准号: 227089-2009
• 负责人: Li, Deli
• 金额: $ 2.08万
• 依托单位: Lakehead University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=523902
• 关键词: Probability; Limit; Theorems; Statistical; Applications

Many of the best-known results in probability concern the asymptotic behavior of distributions; probability limit theory lies at the heart of probability and statistics. Probability limit theorems are concerned with rates of approximation of various observable processes by theoretically recognizable ones. For example, consider a random sample of n observations selected from a population with a finite variance. The central limit theorem states that, when the sample size n is sufficiently large, the sampling distribution of the sum of the n observations will be approximately normally distributed. Thus the central limit theorem allows us to make inferences on the mean of a distribution based on relatively large samples without having to know the exact form of the sampled population. There is a wide variety of statistical applications of probability limit theorems such as the analysis of large data sets, modelling traffic flow in communication networks, and providing a catalyst toward understanding and discussing the role of biostatistical research in relation to health services and decisions. The main focus of this research proposal will be on investigating the asymptotic behavior in statistical applications pertaining to hierarchical models, L-statistics, U-statistics, resampling methods, and the contemporary multivariate data analysis problems such as the largest entry of a sample correlation matrix, misspecified model, kernel estimator of the regression in a left truncation model, etc. A second focus relates to my long-standing research interest in almost sure and weak convergence of random processes, especially in the law of the iterated logarithm, the laws of large numbers, central limit theorems, probabilities of large or moderate deviations, and precise asymptotics in the classical limit theorems for real-valued or Banach-space-valued random processes. The goal are to develop new methods for proving limit theorems and to investigate statistical applications of these theorems. This provides a beautiful interplay between the theory and applications of Statistics, Probability, and Stochastic Processes. The results related to this proposal will be novel and significant insofar as they will extend, generalize, and refine earlier work in the literature.

概率论中许多最著名的结果都与分布的渐近行为有关；概率极限理论是概率论和统计学的核心。概率极限定理是关于各种可观测过程被理论上可识别的过程逼近的速率。例如，考虑从具有有限方差的总体中选择的n个观察值的随机样本。中心极限定理指出，当样本量n足够大时，n个观测值之和的抽样分布近似为正态分布。因此，中心极限定理允许我们在不知道抽样总体的确切形式的情况下，根据相对较大的样本对分布的平均值进行推断。概率极限定理有各种各样的统计应用，例如对大数据集的分析，对通信网络中的交通流量进行建模，以及为理解和讨论生物统计研究在卫生服务和决策方面的作用提供催化剂。本研究计划的主要重点将是研究与分层模型、l统计、u统计、重采样方法有关的统计应用中的渐近行为，以及当代多元数据分析问题，如样本相关矩阵的最大条目、错误指定模型、回归的核估计在左截断模型中等。第二个重点与我长期以来对随机过程的几乎确定和弱收敛的研究兴趣有关，特别是在迭代对数定律，大数定律，中心极限定理，大或中等偏差的概率，以及实值或banach -空间值随机过程经典极限定理中的精确渐近性。目标是开发证明极限定理的新方法，并研究这些定理的统计应用。这在统计学、概率论和随机过程的理论和应用之间提供了一个美丽的相互作用。与此建议相关的结果将是新颖和重要的，因为它们将扩展，概括和完善文献中的早期工作。