BOUNDS FOR THE DISTRIBUTION OF TWO DIMENSIONAL BINARY SCAN STATISTICS Michael V. Boutsikas and Markos V. Koutras Department of Statistics and Insurance Science, University of Piraeus, Greece. Abstract In the present article we develop some efficient bounds for the distribution function of a two dimensional scan statistic defined on a (double) sequence of iid binary trials. The methodology employed here takes advantage of the connection between the scan statistic problem and a equivalent reliability structure and exploits appropriate techniques of reliability theory to establish tractable bounds for the distribution of the statistic of interest. An asymptotic result is established and a numerical study is carried out to investigate the efficiency of the suggested bounds. Keywords and phrases : scan statistics, reliability bounds, covariance bounds, two-dimensional- r-within-consecutive-k1 × k2-out-of-n1 × n2 system. 1 Introduction Assume that a two dimensional rectangular region R = [0,L 1 ] × [0,L 2 ] is observed and our interest is focused on the patterns in which a certain event E occurs in R. let n 1 ,n 2 be two positive integers and define h i = L i /n i ,i =1, 2. Furthermore, assume that n 1 ,n 2 are large enough so that in each of the n 1 × n 2 rectangular subregions R ij = [(i − 1)h 1 , ih 1 ] × [(j − 1)h 2 , ih 2 ],i =1, 2, ..., n 1 ,j =1, 2, ..., n 2 (1) the event E may occur only once or does not occur at all. A question that comes in naturally within this setup is whether reasonable criteria providing evidence of clustering of the occurrences of E over R could be established. In order to establish a probabilistic model, let us introduce the Bernoulli r.v.’s X ij ,i =1, 2, ..., n 1 ,j = 1, 2, ..., n 2 indicating the occurrence (X ij = 1) or non-occurrence (X ij = 0) of the event E in subregion R ij . We assume that X ij are iid r.v.’s with success probabilities q = P (X ij = 1) = 1 − P (X ij = 0) = 1 − p. (2) For 1 ≤ a ≤ n 1 − k 1 +1, 1 ≤ b ≤ n 2 − k 2 +1 we define S(a, b)= a+k 1 −1 X i=a b+k 2 −1 X j=b X ij (3) to be the number of events in a rectangular region comprised of k 1 × k 2 adjacent subregions R ij ,a ≤ i ≤ a + k 1 − 1,b ≤ j ≤ b + k 2 − 1 (k 1 ,k 2 are positive integers). If S(a, b) exceeds a preassigned value r we shall say that (at least) r events have been clustered within the inspected region. The two 1