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

Abstract

For any collection of exchangeable events $A_1, A_2, \cdots, A_k$ the Bonferroni inequalities are usually stated in the form $\max \{N_0, N_2, \cdots, N_{k_e}\} \leqq \mathbf{P}\{\mathbf{\cup}^k_{i=1} A_i\} \leqq \min \{N_1, N_3, \cdots, N_{k_0}\}$ where $N_0 = 0, k_e(k_0)$ is the largest even (odd) integer $\leqq k$, $N_\nu = \sum^\nu_{\alpha=1} (-1)^{\alpha-1}\binom{k}{\alpha}\mathbf{P}_\alpha \quad (\nu = 1, 2, \cdots, k)$ and $\mathbf{P}_\alpha = \mathbf{P}\{A_{i_1} A_{i_2} \cdots A_{i_\alpha}\}$ for any collection of $\alpha$ events. We may regard $N_\nu$ as being of the $\nu$th degree because it involves $\mathbf{P}_1, \mathbf{P}_2, \cdots, \mathbf{P}_\nu$; 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 $\nu = 2, 3$, 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, $\mathbf{P}_k$, for large $k$ in terms of $\mathbf{P}_1, \mathbf{P}_2, \cdots, \mathbf{P}_\nu$. These are also derived and given explicitly for $\nu = 2, 3,$ and 4. Applications of these inequalities to incomplete Dirichlet Type $I$-integrals and to equi-correlated multivariate normal distributions are indicated.