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- W1871331304 abstract "L'identité de Gumbel établit l'égalité de la somme de Bonferroni Sk,n, k = 0, 1, 2, … , n et du moment binomial d'ordre k de la variable qui compte, dans un ensemble arbitraire de névénements, le nombre Mn d'événements se réalisant. Nous présentons un traitement unifié de bornes bien connues obtenues dans ce contexte par Bonferroni, Galambos-Rényi, Dawson-Sankoff et Chung-Erdös, ainsi que de quelques bornes moins connues établies par Fréchet et Gumbel. Toutes font intervenir des sommes de Bonferroni. Notre démarche consiste à montrer que ces bornes apparaissent dans un cadre plus général comme les moments binomiaux d'une variable aléatoire à valeurs entières particulière. L'application de l'identité de Gumbel fournit alors la forme usuelle en termes de sommes de Bonferroni. Notre approche simplifie les preuves existantes, et permet d'étendre les résultats de Fréchet et Gumbel au cas de la probabilité pour qu'au moins t, 1 ≤t≤n des névénements considérés se réalisent. Une dernière conséquence de notre approche est l'amélioration d'une borne de Petrov qui elle-même est la généralisation de la borne de Chung et Erdös. Gumbel’s Identity equates the Bonferroni sum with the k -th binomial moment of the number of eventsMnwhich 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." @default.
- W1871331304 created "2016-06-24" @default.
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- W1871331304 date "2012-05-28" @default.
- W1871331304 modified "2023-09-23" @default.
- W1871331304 title "Gumbel’s Identity, Binomial Moments, and Bonferroni Sums" @default.
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- W1871331304 doi "https://doi.org/10.1111/j.1751-5823.2011.00174.x" @default.
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