Combination setwise-Bonferroni-type bounds

We consider three classes of lower bounds to P(c) = P(X 1 ≤c 1 ,...,X n ≤c); Bonferroni-type bounds, product-type bounds and setwise bounds. Setwise probability inequalities are shown to be a compromise between product-type and Bonferroni-type probability inequalities. Bonferroni-type inequalities always hold. Product-type inequalities require positive dependence conditions, but are superior to the Bonferroni-type and setwise bounds when these conditions are satisfied. Setwise inequalities require less stringent positive dependence bound conditions than the product-type bounds. Neither setwise nor Bonferroni-type bounds dominate the other. Optimized setwise bounds are developed. Results pertaining to the nesting of setwise bounds are obtained. Combination setwise-Bonferroni-type bounds are developed in which high dimensional setwise bounds are applied and second and third order Bonferroni-type bounds are applied within each subvector of the setwise bounds. These new combination bounds, which are applicable for associated random variables, are shown to be superior to Bonferroni-type and setwise bounds for moving averages and runs probabilities. Recently proposed upper bounds to P(c) are reviewed. The lower and upper bounds are tabulated for various classes of multivariate normal distributions with banded covariance matrices. The bounds are shown to be surprisingly accurate and are much easier to compute than the inclusion-exclusion bounds. A strategy for employing the bounds is developed.