Let \(A_1,A_2,\dots,A_n\) be events and let \(m_n\) be the number of those \(A_j\) which occur. A Bonferroni-type inequality is an estimate on the distribution of \(m_n\) by linear combinations of the binomial moments of \(m_n\). When the binomial moments of \(m_n\) are replaced by a selected number of the probabilities of intersections, we speak of restrictive sieves or inequalities related to Bonferroni-types.
The first such inequalities were obtained byA. Rényi [Magyar Tud. Akad. Mat. Fiz. Tud. Oszt. Közl. 11, 79–105 (1961;Zbl 0141.16504)] andA. Rényi and the reviewer [Stud. Sci. Math. Hungar. 3, 351–358 (1968;Zbl 0162.48902)], who make the selection of intersections in the estimates by an arbitrary graph. Independently, selection via a tree was proposed byD. Hunter [J. Appl. Probab. 13, 597–603 (1976;Zbl 0349.60007)], which was rediscovered byK. J. Worsley [Biometrika 69, 297–302 (1982;Zbl 0497.62027)]. All these restrictive inequalities are valid on any probability space and whatever the events \(A_j\) are.
The authors here propose a deviation from such general inequalities and establish restrictive sieves which are valid only when the \(A_j\) satisfy some structural restrictions. They show that such structures naturally appear in some problems of statistical inference and to such situations applications of the new inequalities are given.

Reviewer: Janos Galambos (Philadelphia)