Mathematical Sciences: Probability, Statistics and Functional Analysis

首席研究员将继续调查Vapnik-Cervonenkis(V-C)类在多变量分布估计中的使用。估计值的有效性是根据某一区域C集合的估计值与真实概率的偏差来判断的。如果C类打碎了不包含m个点的集合，则它称为m的V-C类。根据定义，如果F的每个子集都可以被视为F与C的某个元素的交集，则C类粉碎集合F。主要研究者以前已经证明，只有当C是V-C类时，估计值与真实概率乘以n的平方根的偏差才接近所有可能分布的正态极限。通过对固定n的偏差找到严格的界并设计计算有效的方法，这一理论结果将变得实用和有用。其他应用涉及正态和分类问题的测试。

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.