Journal ArticleDOI
Rates of growth and sample moduli for weighted empirical processes indexed by sets
TLDR
Probability inequalities for the supremum of a weighted empirical process indexed by a Vapnik-Cervonenkis class C of sets were obtained in this article for the assumption P(∪{C∈C:P(C)<t})»0 as t»0.Abstract:
Probability inequalities are obtained for the supremum of a weighted empirical process indexed by a Vapnik-Cervonenkis class C of sets. These inequalities are particularly useful under the assumption P(∪{C∈C:P(C)<t})»0 as t»0. They are used to obtain almost sure bounds on the rate of growth of the process as the sample size approaches infinity, to find an asymptotic sample modulus for the unweighted empirical process, and to study the ratio Pn/P of the empirical measure to the actual measure.read more
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References
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Classification and regression trees
TL;DR: The methodology used to construct tree structured rules is the focus of a monograph as mentioned in this paper, covering the use of trees as a data analysis method, and in a more mathematical framework, proving some of their fundamental properties.
Book ChapterDOI
Probability Inequalities for sums of Bounded Random Variables
TL;DR: In this article, upper bounds for the probability that the sum S of n independent random variables exceeds its mean ES by a positive number nt are derived for certain sums of dependent random variables such as U statistics.
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On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities
TL;DR: This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady.
Book
Convergence of stochastic processes
TL;DR: In this paper, the authors define a functional on Stochastic Processes as random functions and propose a uniform convergence of empirical measures in Euclidean spaces, based on the notion of convergence in distribution.