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

Author

Listed:
  • Yi-Shen Lin

    (Yancheng Teachers University)

  • Xenos Chang-Shuo Lin

    (Aletheia University)

  • Daniel Wei-Chung Miao

    (National Taiwan University of Science and Technology)

  • Yi-Ching Yao

    (Academia Sinica)

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.

Suggested Citation

  • Yi-Shen Lin & Xenos Chang-Shuo Lin & Daniel Wei-Chung Miao & Yi-Ching Yao, 2020. "Corrected Discrete Approximations for Multiple Window Scan Statistics of One-Dimensional Poisson Processes," Methodology and Computing in Applied Probability, Springer, vol. 22(1), pages 237-265, March.
    • Handle: RePEc:spr:metcap:v:22:y:2020:i:1:d:10.1007_s11009-019-09704-w
    DOI: 10.1007/s11009-019-09704-w
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    References listed on IDEAS

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    1. Janson, Svante, 1984. "Bounds on the distributions of extremal values of a scanning process," Stochastic Processes and their Applications, Elsevier, vol. 18(2), pages 313-328, November.
    2. Chan, Hock Peng & Zhang, Nancy Ruonan, 2007. "Scan Statistics With Weighted Observations," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 595-602, June.
    3. Joseph I. Naus & Sylvan Wallenstein, 2004. "Multiple Window and Cluster Size Scan Procedures," Methodology and Computing in Applied Probability, Springer, vol. 6(4), pages 389-400, December.
    4. M. Koutras & V. Alexandrou, 1995. "Runs, scans and URN model distributions: A unified Markov chain approach," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 47(4), pages 743-766, December.
    5. Joseph Glaz & Zhenkui Zhang, 2004. "Multiple Window Discrete Scan Statistics," Journal of Applied Statistics, Taylor & Francis Journals, vol. 31(8), pages 967-980.
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