Loader, Clive R. Large-deviation approximations to the distribution of scan statistics. (English) Zbl 0741.60036 Adv. Appl. Probab. 23, No. 4, 751-771 (1991). The paper (which forms part of the author’s Ph.D. dissertation (Stanford University)) deals with three distinct but related topics in the area of testing for homogeneity of \(d\)-dimensional (\(d=1,2\)) Poisson processes.1. The usual scan test statistic \(M_ \Delta\) (maximal number of events in a window of fixed, known length \(\Delta\) which is moved across the unit interval) in a likelihood ratio testing situation has a known but computationally difficult distribution. Computationally simple large deviation approximations for \(M_ \Delta\) (conditional upon a total of \(n\) events in \([0,1]\)) involving the binomial distribution are given for the case \(\Delta\) known. Complete proofs of these results are provided. Furthermore, a performance table comparing the exact and the new approximate tail probabilities for \(M_ \Delta\) is provided (for \(\Delta= .5\) only).2. The scan statistic in (1) relies on the often unrealistic assumption that \(\Delta\) is known. A generalized scan statistic which allows \(\Delta\) to vary is proposed. Approximate distributions are obtained using large-deviation scaling of the resulting boundary-crossing probabilities. Explicit proofs relying on the first-passage time methodology are given. Again a performance table is provided, this time comparing the large deviation and a Gaussian approximation to simulation results.3. The two-dimensional scan statistic \(M_{\Delta_ 1\Delta_ 2}\) for detecting clustering or non-homogeneity can be defined for known \(\Delta_ 1\) and \(\Delta_ 2\) in a fashion similar to that used in the one-dimensional case, replacing the one-dimensional window of length \(\Delta\) by a rectangular window. The fact that rectangular windows are used is central to the proof of the large deviation approximation which relies on a decomposition of the resulting random field into independent one-dimensional processes. Creating an artificial ordering which allows the definition of first-passage times and applying the superposition method, a large deviation result for the distribution of \(M_{\Delta_ 1\Delta_ 2}\) (conditional upon a total of \(n\) events) and the corresponding performance table for the special case \(\Delta_ 1=\Delta_ 2= .5\) are presented. Extensions to the case \(\Delta_ 1\) and \(\Delta_ 2\) unknown are also discussed. Reviewer: V.R.Eastwood (Wolfville) Cited in 30 Documents MSC: 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60D05 Geometric probability and stochastic geometry Keywords:Poisson processes; scan test statistic; large deviation; Gaussian approximation; random field PDFBibTeX XMLCite \textit{C. R. Loader}, Adv. Appl. Probab. 23, No. 4, 751--771 (1991; Zbl 0741.60036) Full Text: DOI