On the limiting distribution of the spatial scan statistic

Bootstrap is the standard method in the spatial scan test. However, because the spatial scan statistic lacks theoretical properties, its development and connection to mainstream statistics has been limited. Using the methods of empirical processes with a few weak regularity conditions, the limiting distribution of the spatial scan statistic, which can provide a theoretical basis for the spatial scan test, is derived. It is shown that the limiting distribution of the spatial scan statistic only depends on the ratio of at risk populations and the collection of cluster candidates, which provides a base to theoretically assess the critical value of the spatial scan test in a real world daily or weekly disease surveillance. A simulation study based on the Kolmogorov-Smirnov test shows that the limiting distribution is consistent with the true distribution. Type I error probabilities and power functions from the limiting distribution and the bootstrap method are almost identical.