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Asymptotic properties of spatial scan statistics under the alternative hypothesis

A common challenge for most spatial cluster detection methods is the lack of asymptotic properties to sup-port their validity. As the spatial scan test is the most often used cluster detection method, we investigate two important properties in the method: the consistency and asymptotic local efﬁciency. We address the consistency by showing that the detected cluster converges to the true cluster in probability. We address the asymptotic local efﬁciency by showing that the spatial scan statistic asymptotically converges to the square of the maximum of a Gaussian random ﬁeld, where the mean and covariance functions of the Gaussian random ﬁeld depends on a function of at-risk population within and outside of the cluster. These conclusions, which are also supported by simulation and case studies, make it practical to precisely detect and characterize a spatial cluster.