Gumbel's Identity, Binomial Moments, and Bonferroni Sums

Abstract

Gumbel's Identity equates the Bonferroni sum with the k -th binomial moment of the number of eventsM nwhich occur, out ofnarbitrary events. We provide a unified treatment of familiar probability bounds on a union of events by Bonferroni, Galambos-Rényi, Dawson-Sankoff, and Chung-Erdös, as well as less familiar bounds by Fréchet and Gumbel, all of which are expressed in terms of Bonferroni sums, by showing that all these arise as bounds in a more general setting in terms of binomial moments of a general non-negative integer-valued random variable. Use of Gumbel's Identity then gives the inequalities in familiar Bonferroni sum form. This approach simplifies existing proofs. It also allows generalization of the results of Fréchet and Gumbel to give bounds on the probability that at leasttofnevents occur for anyA further consequence of the approach is an improvement of a recent bound of Petrov which itself generalizes the Chung-Erdös bound. © 2012 The Authors. International Statistical Review © 2012 International Statistical Institute.