Approximations for a Three Dimensional Scan Statistic

Let X ijk ,1 ≤ i ≤ N 1 ,1 ≤ j ≤ N 2 , 1 ≤ k ≤ N 3 be a sequence of independent and identically distributed 0 − 1 Bernoulli trials. X ijk = 1 if an event has occurred at the i , j , k th location in a three dimensional rectangular region and X ijk = 0, otherwise. For 2 ≤ m j ≤ N j − 1,1 ≤ j ≤ 3, a three dimensional discrete scan statistic is defined as the maximum number of events in any m 1 × m 2 × m 3 rectangular sub-region in the entire N 1 × N 2 × N 3 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.