Hoeffding Inequality Research Articles

On the maximal inequality

In this note, we establish a sequence of maximal inequalities for sums of i.i.d. random variables which sharpen Hoeffding's inequality and many other similar results.

A Combined Bound for Errors in Auditing Based on Hoeffding's Inequality and the Bootstrap

For an accounting population's total error amount, confidence bounds that are reliable but not too conservative help auditors avoid costly overauditing. Parametric methods that have been proposed are not always robust. To date, none of the nonparametric methods proposed have shown superiority over the others in all population settings. This article describes a simulation study of a nonparametric bound on the total error amount in an auditing population when dollar-unit sampling is employed. The bound is a combination of two new bounds, one generated by Hoeffding's inequality and the other by a modified nonparametric bootstrap-t method that makes use of today's powerful yet cheap and readily available computer technology. At the 95% confidence level for sample size 120, the combined bound is reliable over a wide range of simulated study populations and is on average 13% smaller than the benchmark Stringer bound based on the binomial distribution. For real accounting populations, the combined bound remains ...

A Combined Bound for Errors in Auditing Based on Hoeffding's Inequality and the Bootstrap

A Combined Bound for Errors in Auditing Based on Hoeffding's Inequality and the Bootstrap

Hoeffding's inequalities: A counterexample

A sequence ψ: ℕ0 → ℝ satisfiesHoeffding's inequality of order n if wheneverX1,...,Xn are independent nonnegative integer-valued elementary random variables and\(\tilde X_1 , \ldots ,\tilde X_n \) are independent identically distributed nonnegative integer-valued elementary random variables, the common distribution of which is the average of those ofX1,...,Xn. We show that for each integerm greater than 2 there exists a sequence ψ satisfying Hoeffding's inequality of every order greater thanm but not that of orderm. This answers a question raised by Berg, Christensen, and Ressel.

Probability inequalities for continuous unimodal random variables with finite support

A paramecer-free Bernstein-type upper bound is derived for the probability that the sum S of n i.i.d, unimodal random variables with finite support, X1 ,X2,…,Xn, exceeds its mean E(S) by the positive value nt. The bound for P{S - nμ ≥ nt} depends on the range of the summands, the sample size n, the positive number t, and the type of unimodality assumed for Xi. A two-sided Gauss-type probability inequality for sums of strongly unimodal random variables is also given. The new bounds are contrasted to Hoeffding's inequality for bounded random variables and to the Bienayme-Chebyshev inequality. Finally, the new inequalities are applied to a classic probability inequality example first published by Savage (1961).

A Note on Hoeffding's Inequality

Abstract A lower bound is derived concerning a special form of the entropy function of information theory. It is applied to Hoeffding's bound for the probability of the deviation of the sample mean from its expected value and for the corresponding problem concerning the sample variance. Furthermore, applying the bound one can obtain a discard function for a binary symmetric channel, a lower bound for the total error probability in testing two simple hypotheses, given the number of observations, and a lower bound for the deviation of the empirical distribution from the true distribution.

A Note on Hoeffding's Inequality

Abstract A lower bound is derived concerning a special form of the entropy function of information theory. It is applied to Hoeffding's bound for the probability of the deviation of the sample mean from its expected value and for the corresponding problem concerning the sample variance. Furthermore, applying the bound one can obtain a discard function for a binary symmetric channel, a lower bound for the total error probability in testing two simple hypotheses, given the number of observations, and a lower bound for the deviation of the empirical distribution from the true distribution.