Approximations for a conditional two-dimensional scan statistic

Let Xi,j,1[less-than-or-equals, slant]i[less-than-or-equals, slant]n1,1[less-than-or-equals, slant]j[less-than-or-equals, slant]n2, be a sequence of independent and identically distributed nonnegative integer valued random variables. The observation Xi,j denotes the number of events that have occurred in the i,jth location in a two dimensional rectangular region R. For 2[less-than-or-equals, slant]mi[less-than-or-equals, slant]ni-1, i=1,2, the two dimensional discrete scan statistic is defined as the maximum number of events in any of the m1 by m2 consecutive rectangular windows in that region. Conditional on the total number of events that have occurred in R, we refer to this scan statistic as the conditional two-dimensional scan statistic. Two-dimensional scan statistics have been extensively used in many areas of science to analyze the occurrence of observed clusters of events in space. Since this scan statistic is based on highly dependent consecutive subsequences of observed data, accurate approximations for its distributions are of great value. In this article, based on the scanning window representation of the scan statistic, accurate product-type, Poisson and a compound Poisson approximations are investigated. Moreover, accurate approximations for the expected size of the scan statistic are derived. Numerical results are presented to evaluate the performance of the approximations discussed in this article.