Large Sample Problems Arising on the Borderline between Probability and Mathematical Statistics

概率论与数理统计边界出现的大样本问题

• 批准号: 0503908
• 负责人: David Mason
• 金额: --
• 依托单位: University of Delaware
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-08-01 至 2011-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0503908&HistoricalAwards=false
• 关键词: Large; Sample; Problems; Arising; Borderline

The investigator plans to concentrate his efforts on large sample theoryproblems connected with self-normalized sums and functionals of the empiricaldistribution. His interest in self-normalized sums arose largely out of thequestion: When is the Student-t statistic asymptotically normal? Among hisgoals in this area, he hopes to complete the theory of the asymptoticdistribution of self-normalized sums. The particular functionals of theempirical distribution that he plans to investigate include statistics andestimators based on distance measures such as the empirical Hellinger distanceand a class of integral functionals of density estimators, which are used toestimate entropy and the integral of the squared density and its derivative.This will likely lead to the development of a theory of local U-statisticsprocesses. This theory would be analogous to the local empirical process theorythat he has worked out with a number of his long-term collaborators to studythe large sample properties of kernel-type function estimators. These includekernel-type estimators of the density function, the regression function and theconditional distribution function. Local empirical process theory has shownitself to be a very powerful tool to determine exact rates of point-wise anduniform consistency of kernel-type nonparametric function estimators. A localU-statistics process theory should be at least as applicable. The researchproblems outlined in this proposal should lead to the creation of a number ofnew and useful methods to treat large sample theory problems.The investigator works in the borderline area between probability theory andmathematical statistics. He is delighted when he uncovers unexpectedprobabilistic phenomena that occasionally appear on this frontier. The problemsthat he is interested in are frequently motivated by questions about the limitbehavior of statistical tests and estimators. They arise whenever one wants tomake a decision on the basis of limited knowledge. This is especially the casewhen it is not feasible to sample an entire population to settle a questionabout that population. Some typical examples are when a medical investigator isinterested in deciding whether or not a proposed new drug or medical treatmentis effective or a fisheries scientist wants to estimate the number of salmonthat will return this season to spawn.

研究者计划集中精力于大样本理论问题与自正规化和泛函的degeneraldistribution。他的兴趣在自我正规化的总和产生了很大程度上的问题：当是学生t统计渐近正常？在他的目标在这一领域，他希望完成理论的渐近分布的自我规范化的总和。他计划研究的经验分布的特殊泛函包括基于距离测度的统计和估计，如经验Hellinger距离和一类用于估计熵和平方密度及其导数的积分的密度估计的积分泛函。这将可能导致局部U-微分过程理论的发展。这一理论将类似于当地的经验过程theorythat他已经制定了一些他的长期合作者研究大样本性质的核函数估计。这些估计包括密度函数、回归函数和一致性分布函数的核型估计。局部经验过程理论已被证明是确定核型非参数函数估计的逐点相合和一致相合的精确速度的一个非常有力的工具。局部U统计过程理论至少应该是适用的。在这个建议中概述的研究问题应该导致创造一些新的和有用的方法来处理大样本的理论问题。调查工作在概率论和数理统计之间的边界区域。当他发现偶尔出现在这一前沿领域的意外概率现象时，他很高兴。他感兴趣的问题经常是由关于统计检验和估计的极限行为的问题所激发的。每当一个人想在有限的知识基础上做出决定时，它们就会出现。 尤其是当对整个人口进行抽样以解决关于该人口的问题是不可行的时候。一些典型的例子是，当一个医学调查员有兴趣决定一种新的药物或医疗方法是否有效，或者一个渔业科学家想估计这个季节会返回产卵的鲑鱼数量。