DEGREE, ITERATION AND PERMUTATION IN IMPROVING BONFERRONI‐TYPE BOUNDS

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Abstract
Summary: This note has as its starting points: 1) the upper bound of Worsley (1982) for the probability of union of n events: 2) the iterative procedure of Hoppe (1985) for deriving successive upper and lower Bonferroni‐type bounds, in alternation, for the probability of such a union. We first show that Worsley's bound (and hence successive bounds based on it) can be sharpened by an initial application of 2) to an appropriate lower bound due to Boole (1854). Subsequently, we show that if the n events are exchangeable, the sequence of bounds obtained from this starting point is that due to Sobel and Uppuluri (1972). The role of degree of a bound is explored, as well as sharpening of bounds by permutation when events are not necessarily exchangeable.

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