Approximation and simulation of the distributions of scan statistics for Poisson processes in higher dimensions (Q1294764)

Clustering of points in Poisson processes, in one or more dimensions, is of interest in many applications, for example, in risk analysis, telecommunication, epidemiology, reliability, and traffic theory. In risk analysis, one of the objects is to study the risk of collapse of a structure due to an extreme load. This load could occur during the normal load, but may also be caused by other circumstances. The author is mainly interested in examining the distribution of the extreme load during the normal load variation over a period of time. Given a Poisson process in two or three dimensions, he is interested in the scan statistic, i.e. the largest number of points contained in a translate of a fixed scanning set restricted to lie inside a rectangular area. The distribution of the scan statistic is accurately approximated for rectangular scanning sets using a technique that is also extended to the higher dimensions. The accuracy of the approximation is checked through simulation. Throughout the paper, the scan statistics of homogeneous Poisson processes are studied.