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

Milton Sobel and V. R. R. Uppuluri
The Annals of Mathematical Statistics
Vol. 43, No. 5 (Oct., 1972), pp. 1549-1558
Stable URL: http://www.jstor.org/stable/2240077
Page Count: 10
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
On Bonferroni-Type Inequalities of the Same Degree for the Probability of Unions and Intersections
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Abstract

For any collection of exchangeable events A1, A2, ⋯, Ak the Bonferroni inequalities are usually stated in the form max {N0, N2, ⋯, Nke } ≤ P{∪k i=1 Ai} ≤ 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 Nν as 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.

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