On the probability of large deviations from the expectation for sums of bounded, independent random variables

Bennett, G. G.

doi:10.1093/biomet/50.3-4.528

Cited by 26 publications

(4 citation statements)

References 5 publications

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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%

Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

Shilane

Bean

2013

Journal of Probability and Statistics

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The negative binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard normal approximation often does not provide adequate inferences about the data's expected value in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's inequality. We now propose growth estimators of the negative binomial mean. Under high dispersion, zero values are likely to be overrepresented in the data. A growth estimator constructs a normal-style confidence interval by effectively removing a small, predetermined number of zeros from the data. We propose growth estimators based upon multiplicative adjustments of the sample mean and direct removal of zeros from the sample. These methods do not require estimating the nuisance dispersion parameter. We will demonstrate that the growth estimators' confidence intervals provide improved coverage over a wide range of parameter values and asymptotically converge to the sample mean. Interestingly, the proposed methods succeed despite adding both bias and variance to the normal approximation.

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Section: Bernstein's Inequalitymentioning

confidence: 99%

Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

Shilane

Bean

2013

Journal of Probability and Statistics

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“…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%

On Bernstein Type Exponential Inequalities for Matrix Martingales

Tian¹

2021

Preprint

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In this work, Bernstein's concentration inequalities for squared integrable matrix-valued discrete-time martingales are obtained. Based on Lieb's theory and Bernstein's condition, a suitable supermartingale can be constructed.Our proof is largely based on this new exponential supermartingale, Freedman's method, and Doob's stopping theorem. Our result can be regarded as an extension of Tropp's work (ECP, 2012).

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“…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%

Section: Introductionmentioning

confidence: 99%

On Bernstein type inequalities for stochastic integrals of multivariate point processes

Wang

Lin

2019

Stochastic Processes and their Applications

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We consider the stochastic integrals of multivariate point processes and study their concentration phenomena. In particular, we obtain a Bernstein type of concentration inequality through Doléans-Dade exponential formula and a uniform exponential inequality using a generic chaining argument. As applications, we obtain a upper bound for a sequence of discrete time martingales indexed by a class of functionals, and so derive the rate of convergence for nonparametric maximum likelihood estimators, which is an improvement of earlier work of van de Geer.

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On the probability of large deviations from the expectation for sums of bounded, independent random variables

Cited by 26 publications

References 5 publications

Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

Growth Estimators and Confidence Intervals for the Mean of Negative Binomial Random Variables with Unknown Dispersion

On Bernstein Type Exponential Inequalities for Matrix Martingales

On Bernstein type inequalities for stochastic integrals of multivariate point processes

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