Approximations and Bounds for the Distribution of the Scan Statistic

Abstract

Abstract The scan statistic is used in many areas of science to test the null hypothesis of uniformity against a clustering alternative. This article derives approximations and bounds for the distribution of scan statistics. An extensive simulation study is carried out to compare the approximations and the bounds derived in this article with other known approximations and bounds. The new lower and upper bounds that have been derived are very useful in evaluating the approximations and the simulation. A quantity of average number of points in the scanning interval of length d, μd , is introduced to study the approximations for the distribution of the scan statistic. For low values of μd the approximation derived in this article is the most accurate one. The importance of the scan statistics arises from their applications in many areas, including nuclear physics, geology, radio-optics, photography, and epidemiology.