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

For any collection of exchangeable events A₁, A₂, ⋯, Aₖ the Bonferroni inequalities are usually stated in the form max ₀, N₂, ⋯, Nkₑ\ P\∪ᵏᵢ₌₁ Aᵢ\ min ₁, N₃, ⋯, Nk₀\ where N₀ = 0, kₑ(k₀) is the largest even (odd) integer k, N_ν = ∑^να=1 (-1)α-1kαP_α (ν = 1, 2, ⋯, k) and P_α = P₁ Ai₂ ⋯ Ai_α\ for any collection of α events. We may regard N_ν as being of the νth degree because it involves P₁, P₂, ⋯, P_ν; 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, ⋯, P_ν. 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.