COMBINATION SETWISE-BONFERRONI-TYPE BOUNDS

We consider three classes of lower bounds to P(c) = P(X(1) less than o r equal to c(1),...,X(n) less than or equal to c); Bonferroni-type bou nds, product-type bounds and setwise bounds. Setwise probability inequ alities are shown to be a compromise between product-type and Bonferro ni-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 thes e conditions are satisfied. Setwise inequalities require less stringen t positive dependence bound conditions than the product-type bounds. N either setwise nor Bonferroni-type bounds dominate the other. Optimize d setwise bounds are developed. Results pertaining to the nesting of s etwise bounds are obtained. Combination setwise-Bonferroni-type bounds are developed in which high dimensional setwise bounds are applied an d second and third order Bonferroni-type bounds are applied within eac h subvector of the setwise bounds. These new combination bounds, which are applicable for associated random variables, are shown to be super ior to Bonferroni-type and setwise bounds for moving averages and runs probabilities. Recently proposed upper bounds to P(c) are reviewed. T he lower and upper bounds are tabulated for various classes of multiva riate 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 boun ds is developed. (C) 1996 John Wiley & Sons, Inc.