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

概率和统计学中的大样本问题涉及到当样本量增加到无穷大时，过程或统计数据会发生什么变化。调查人员计划在三个自然会出现此类问题的领域开展工作。这些是：1。自归一化和，2。广义极值，和3。加权近似。区域1中的问题源于一个长期存在的问题，即经典学生t统计量的可能渐近分布。区域2的问题源于极端值的极限分布问题，如年最高温度、河流高度和人口的最大收入。他将极值的概念推广到多变量环境，并提出研究其大样本性质。最后，在区域3中，他打算研究逼近有限样本过程的方法，这些过程是由渐近过程中数据依赖性的检验产生的。这将提供一个有用的工具来确定这种有限样本过程的许多函数的极限分布。这三个研究领域的问题的解决方案应该在统计推断和估计中有大量的应用，特别是在多变量和依赖的情况下，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。无论如何，它们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.