On Bonferroni-Type Inequalities of the Same Degree for the Probability of Unions and Intersections

• Published in: The Annals of mathematical statistics 1972-10, Vol.43 (5), p.1549-1558
• Main Authors: Sobel, Milton, Uppuluri, V. R. R.
• Format: Article
• Language: English
• Subjects: Integers; Mathematical inequalities; Probabilities; Random variables
• ISSN: 0003-4851; 2168-8990
• DOI: 10.1214/aoms/1177692387

For any collection of exchangeable events A1, A2, ⋯, Akthe Bonferroni inequalities are usually stated in the form max {N0, N2, ⋯, Nke } ≤ P{∪k i=1Ai} ≤ min {N1, N3, ⋯, Nk0 } where N0= 0, ke(k0) is the largest even (odd) integer ≤ k,$N_\nu = \sum^\nu_{\alpha=1} (-1)^{\alpha-1}\binom{k}{\alpha}\mathbf{P}_\alpha \quad (\nu = 1, 2, \cdots, k)$and Pα= P{Ai1 Ai2 ⋯ Aiα } for any collection of α events. We may regard Nνas being of the νth degree because it involves P1, P2, ⋯, Pν; hence the lower and upper bounds above are never of the same degree. In this paper we develop improved lower and upper bounds of the same degree. For degree ν = 2, 3, and 4 these results are given explicitly. A related problem is to get lower and upper bounds for the probability of the intersection of events, Pk, for large k in terms of P1, P2, ⋯, Pν. These are also derived and given explicitly for ν = 2, 3, and 4. Applications of these inequalities to incomplete Dirichlet Type I-integrals and to equi-correlated multivariate normal distributions are indicated.