Mathematical Sciences: Probability, Statistics and Functional Analysis

首席研究员将继续他在多元分布估计中使用Vapnik-Cervonenkis （V-C）类的研究。估计的有效性是通过估计与某些区域C集合的真实概率的偏差来判断的。如果一个类C没有“粉碎”m个点的集合，则称为m的V-C类。根据定义，如果F的每个子集都可以看作是F与C的某些元素的交集，则类C会破坏集合F。首席研究员之前已经证明，只有当C是V-C类时，估计与真实概率的偏差乘以根号n，才会接近所有可能分布的正态极限。这一理论结果将通过寻找固定n的偏差的紧界和设计有效的计算方法而变得实用和有用。其他应用涉及正态性和分类问题的测试。

The principal investigator will continue his investigation of the use of Vapnik-Cervonenkis (V-C) classes in estimation of multivariate distributions. The validity of the estimates is judged from the deviation of the estimated from the true probability for some collection of regions C. A class C is called a V-C class for m if it "shatters" no set of m points. By definition a class C shatters a set F if every subset of F can be viewed as the intersection of F with some element of C. The principal investigator has shown before that the deviation of the estimates from the true probability multiplied by square root of n, approaches a normal limit for all possible distributions only if C is a V-C class. This theoretical result will be made practical and useful by finding tight bounds on the deviations for fixed n and designing computationally efficient methods. Other applications relate to tests for normality and classification problems.