×

Large-deviation approximations to the distribution of scan statistics. (English) Zbl 0741.60036

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.

MSC:

60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
60D05 Geometric probability and stochastic geometry
PDFBibTeX XMLCite
Full Text: DOI