Selected Large Sample Problems in Self-Normalized Sums, Generalized Extremes and Weighted Approximations

9803344梅森概率统计中的大样本问题涉及当样本大小增加到无穷大时，过程或统计会发生什么。调查人员计划在三个自然出现此类问题的领域开展工作。它们如下：1.自归一化和，2.广义极值，3.加权逼近。区域1中的问题源于一个长期存在的问题，即经典的学生t统计量可能的渐近分布。区域2中的问题源于极值的极限分布问题，如年最高气温、河流高度和人口中最高收入。他将极值的概念推广到多变量环境中，并建议研究其大样本性质。最后，在区域3中，他打算研究通过渐近过程检验数据相关性而产生的逼近有限样本过程的方法。这将为确定这类有限样本过程的许多泛函的极限分布提供一个有用的工具。对所有三个研究领域的问题的解决在统计推断和估计中应有许多应用，特别是在多变量和相关情况下，只要有大样本可用。最终，希望许多结果将导致数据分析工具的构建，这些工具将进入统计的日常实践。在任何情况下，它们都应该提高我们对一系列有趣的随机行为的理解。

9803344 Mason Large sample problems in probability and statistics concern what happens to processes or statistics when the sample size increases to infinity. The investigator plans to work in three areas in which such problems naturally arise. These are the following: 1. self-normalized sums, 2. generalized extremes, and 3. weighted approximations. The problems in area 1 originate from the long standing question concerning the possible asymptotic distributions of the classical Student's t-statistic. Those in area 2 stem from questions about the limiting distribution of extreme values, such as annual maximum temperatures, river heights and largest incomes in a population. He generalizes the notion of extremes to a multivariate setting and proposes to study its large sample properties. Finally in area 3, he intends to look into methods of approximating finite sample processes that arise from testing for dependence in data by an asymptotic process. This should provide a useful tool to determine the limiting distribution of many functionals of such finite sample processes. Solutions to problems in all three areas of research should have numerous applications in statistical inference and estimation, especially in multivariate and dependent situations, whenever large samples are available. Eventually, it is hoped that many of the results will lead to the construction of data analytic tools which will enter into the everyday practice of statistics. In any case they should improve our understanding of an interesting range of random behavior.