Abstract
Let Xijk,1 ≤ i ≤ N1,1 ≤ j ≤ N2, 1 ≤ k ≤ N3 be a sequence of independent and identically distributed 0 − 1 Bernoulli trials. Xijk = 1 if an event has occurred at the i,j,kth location in a three dimensional rectangular region and Xijk = 0, otherwise. For 2 ≤ mj ≤ Nj − 1,1 ≤ j ≤ 3, a three dimensional discrete scan statistic is defined as the maximum number of events in any m1×m2×m3 rectangular sub-region in the entire N1×N2×N3 rectangular region. In this article, a product-type approximation and three Poisson approximations are derived for the distribution of this three dimensional scan statistic. Numerical results are presented to evaluate the accuracy of these approximations and their use in testing for randomness.
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