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1 results

  • An upper bound for the probability of a union

  • David Hunter
  • Journal:
    Journal of Applied Probability / Volume 13 / Issue 3 / September 1976
    Published online by Cambridge University Press:
    14 July 2016, pp. 597-603
    Print publication:
    September 1976

  • The problem of bounding P(∪ Ai) given P(Ai) and P(AiAj) for i ≠ j = 1, …, k goes back to Boole (1854) and Bonferroni (1936). In this paper a new family of upper bounds is derived using results in graph theory. This family contains the bound of Kounias (1968), and the smallest upper bound in the family for a given application is easily derivable via the minimal spanning tree algorithm of Kruskal (1956). The properties of the algorithm and of the multivariate normal and t distributions are shown to provide considerable simplifications when approximating tail probabilities of maxima from these distributions.