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

研究者计划将精力集中在与经验分布的自归一化和和泛函相关的大样本理论问题上。他对自归一化和的兴趣很大程度上源于一个问题：Student-t统计量何时渐近正态？在他的目标中，他希望完成自归一化和的渐近分布理论。他计划研究的经验分布的特殊函数包括基于距离度量的统计和估计，如经验海灵格距离和一类密度估计的积分函数，用于估计熵和密度平方及其导数的积分。这可能会导致局部u统计过程理论的发展。这个理论类似于局部经验过程理论，他和他的一些长期合作者一起研究核型函数估计器的大样本性质。这包括密度函数、回归函数和条件分布函数的核估计。局部经验过程理论已经证明自己是一个非常强大的工具，以确定核型非参数函数估计的点方向和一致一致性的精确率。本地统计过程理论至少应该同样适用。在这个建议中概述的研究问题应该导致创建一些新的和有用的方法来处理大样本理论问题。研究者在概率论和数理统计之间的边缘领域工作。当他发现偶尔出现在这一前沿的意外概率现象时，他很高兴。他感兴趣的问题常常是由关于统计测试和估计器的局限性行为的问题引起的。当一个人想在有限的知识基础上做出决定时，就会出现这种情况。当无法对整个人口进行抽样以解决有关该人口的问题时，尤其如此。一些典型的例子是，当一个医学研究者想要确定一个提议的新药或医疗是否有效，或者一个渔业科学家想要估计这个季节将返回产卵的鲑鱼的数量。

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.