1963
DOI: 10.1093/biomet/50.3-4.528
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On the probability of large deviations from the expectation for sums of bounded, independent random variables

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Cited by 26 publications
(4 citation statements)
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“…Tail probability bounds provide an alternative methodology that typically rely upon more mild assumptions about the data. Bounds such as Bernstein's Inequality (Bernstein, 1934), Bennett's Inequality (Bennett, 1962(Bennett, , 1963, or methods based on the work of Hoeffding (1963) and Berry-Esseen (Berry, 1941;Esseen, 1942Esseen, , 1956van Beek, 1972) may be employed. Rosenblum and van der Laan (2009) apply these bounds to produce confidence intervals based on the estimators' empirical influence curves.…”
Section: Bernstein's Inequalitymentioning
confidence: 99%
“…Tail probability bounds provide an alternative methodology that typically rely upon more mild assumptions about the data. Bounds such as Bernstein's Inequality (Bernstein, 1934), Bennett's Inequality (Bennett, 1962(Bennett, , 1963, or methods based on the work of Hoeffding (1963) and Berry-Esseen (Berry, 1941;Esseen, 1942Esseen, , 1956van Beek, 1972) may be employed. Rosenblum and van der Laan (2009) apply these bounds to produce confidence intervals based on the estimators' empirical influence curves.…”
Section: Bernstein's Inequalitymentioning
confidence: 99%
“…As is known to all, a lot of exponential types inequalities are well known and frequently employed in statistics and probability. Especially, when considering the partial sum of independent random variables, there are multiple classical inequalities such as Petrov, Hoeffding [2], Bennett [3,4] and Bernstein [5]. Bernstein's inequality is crucial because it gives an exponential upper bound on the tail probability of a large class of random variables.…”
Section: Introductionmentioning
confidence: 99%
“…for all x > 0 and for all K satisfying V ar(S n ) = n i=1 Eξ 2 n ≤ K. (1.1) was due to Bernstein [6], and so (1.1) is now referred as Bernstein condition. Since then various extensions and improvement have appeared in literature, among which are Bennett [1,2], Hoeffding [9], Freedman [8], Bentkus [4,5], Fan et al [7]. A very recent nice book is Bercu et al [3] which gives a very clear exposition on concentration inequalities for sums of independent random variables and martingales.…”
Section: Introductionmentioning
confidence: 99%
“…for all x > 0 and for all K satisfying V ar(S n ) = n i=1 Eξ 2 n ≤ K. (1.1) was due to Bernstein [6], and so (1.1) is now referred as Bernstein condition. Since then various extensions and improvement have appeared in literature, among which are Bennett [1,2], Hoeffding [9], Freedman [8],…”
Section: Introductionmentioning
confidence: 99%