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- Glivenko–Cantelli_theorem abstract "In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets. The Glivenko–Cantelli classes arise in Vapnik–Chervonenkis theory, with applications to machine learning. Applications can be found in econometrics making use of M-estimators.Assume that are independent and identically-distributed random variables in with common cumulative distribution function . The empirical distribution function for is defined bywhere is the indicator function of the set . For every (fixed) , is a sequence of random variables which converge to almost surely by the strong law of large numbers, that is, converges to pointwise. Glivenko and Cantelli strengthened this result by proving uniform convergence of to .Theoremalmost surely.This theorem originates with Valery Glivenko, and Francesco Cantelli, in 1933.RemarksIf is a stationary ergodic process, then converges almost surely to . The Glivenko–Cantelli theorem gives a stronger mode of convergence than this in the iid case.An even stronger uniform convergence result for the empirical distribution function is available in the form of an extended type of law of the iterated logarithm. See asymptotic properties of the Empirical distribution function for this and related results.↑ ↑ ↑ ↑ ↑ ↑".
- Glivenko–Cantelli_theorem wikiPageID "3828409".
- Glivenko–Cantelli_theorem wikiPageRevisionID "595279185".
- Glivenko–Cantelli_theorem subject Category:Asymptotic_statistical_theory.
- Glivenko–Cantelli_theorem subject Category:Empirical_process.
- Glivenko–Cantelli_theorem subject Category:Probability_theorems.
- Glivenko–Cantelli_theorem subject Category:Statistical_theorems.
- Glivenko–Cantelli_theorem comment "In the theory of probability, the Glivenko–Cantelli theorem, named after Valery Ivanovich Glivenko and Francesco Paolo Cantelli, determines the asymptotic behaviour of the empirical distribution function as the number of independent and identically distributed observations grows. The uniform convergence of more general empirical measures becomes an important property of the Glivenko–Cantelli classes of functions or sets.".
- Glivenko–Cantelli_theorem label "Glivenko–Cantelli theorem".
- Glivenko–Cantelli_theorem label "Gliwenko-Cantelli-Satz".
- Glivenko–Cantelli_theorem label "Stelling van Glivenko–Cantelli".
- Glivenko–Cantelli_theorem label "Teorema di Glivenko-Cantelli".
- Glivenko–Cantelli_theorem label "Théorème de Glivenko-Cantelli".
- Glivenko–Cantelli_theorem label "Теорема Гливенко — Кантелли".
- Glivenko–Cantelli_theorem sameAs Glivenko%E2%80%93Cantelli_theorem.
- Glivenko–Cantelli_theorem sameAs Gliwenko-Cantelli-Satz.
- Glivenko–Cantelli_theorem sameAs Théorème_de_Glivenko-Cantelli.
- Glivenko–Cantelli_theorem sameAs Teorema_di_Glivenko-Cantelli.
- Glivenko–Cantelli_theorem sameAs Stelling_van_Glivenko–Cantelli.
- Glivenko–Cantelli_theorem sameAs Q544369.
- Glivenko–Cantelli_theorem sameAs Q544369.
- Glivenko–Cantelli_theorem wasDerivedFrom Glivenko–Cantelli_theorem?oldid=595279185.