A Fréchet-Optimal Strengthening of the Dawson-Sankoff Lower Bound
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Abstract
This paper proposes a lower bound for the probability that at least one out of \(n\) arbitrary events occurs. The information used consists of the first- and second- degree Bonferroni summations in conjunction with \(p_1\) and \(p_n\), where \(p_1\) is the probability that exactly one event occurs and \(p_n\) is the probability that all \(n\) events occur. We prove that the proposed bound is a Fréchet optimal lower bound, which is a criterion difficult to achieve in general. The two additional non-negative terms used in the proposed bound make it at least as good as the Dawson–Sankoff lower bound, a Fréchet optimal degree two lower bound using the first- and second- degree Bonferroni summations only. A numerical example is presented to illustrate that in some cases, the improvement can be substantial.
Keywords
Binomial moments Fréchet optimality Dawson–Sankoff inequality Bonferroni-type boundsAMS 2000 Subject Classification
Primary 60E15 Secondary 62P10Preview
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