Corrected Discrete Approximations for Multiple Window Scan Statistics of One-Dimensional Poisson Processes

Abstract

In the literature on scan statistics, the distributions of continuous scan statistics for one-dimensional Poisson processes have been extensively studied, most of which deal with single window scan statistics under homogeneous Poisson processes. In this paper, we consider discrete approximations for the distributions of multiple window scan statistics of homogeneous/nonhomogeneous Poisson processes. We derive the first-order terms of the discrete approximations, which involve some functionals of the Poisson processes. We then apply Richardson’s extrapolation to yield corrected (second-order) approximations. Numerical results are presented to show the accuracy of the approximations.