Mathematical Sciences: Probability, Statistics and Functional Analysis

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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.