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Testing proportionality between the first-order intensity functions of spatial point processes

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  • Zhang, Tonglin
  • Zhuang, Run

Abstract

This article proposes a Kolmogorov–Smirnov type test for proportionality between the first-order intensity functions of two independent spatial point processes. After appropriate scaling, the test statistic is constructed by maximizing the absolute difference between their point densities over a π-system. By treating non-stationary point processes as transformed from stationary point processes such that all questions of asymptotics related to the tightness can be answered, the article shows that the resulting test statistic converges weakly to the absolute maximum of a pinned Brownian sheet. This may be reduced to the standard Brownian bridge in a special case. A simulation study shows that the type I error probability of the test is close to the significance level and the power function increases to 1 as the magnitude of non-proportionality increases. In applications to two typical natural hazard data, the article concludes that the first-order intensity functions might be proportional in one case and not in the other.

Suggested Citation

  • Zhang, Tonglin & Zhuang, Run, 2017. "Testing proportionality between the first-order intensity functions of spatial point processes," Journal of Multivariate Analysis, Elsevier, vol. 155(C), pages 72-82.
    • Handle: RePEc:eee:jmvana:v:155:y:2017:i:c:p:72-82
    DOI: 10.1016/j.jmva.2016.11.013
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    References listed on IDEAS

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    1. Yongtao Guan & Michael Sherman & James A. Calvin, 2006. "Assessing Isotropy for Spatial Point Processes," Biometrics, The International Biometric Society, vol. 62(1), pages 119-125, March.
    2. Yosihiko Ogata, 1998. "Space-Time Point-Process Models for Earthquake Occurrences," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 50(2), pages 379-402, June.
    3. Yongtao Guan, 2008. "A KPSS Test for Stationarity for Spatial Point Processes," Biometrics, The International Biometric Society, vol. 64(3), pages 800-806, September.
    4. Ivanoff, Gail, 1982. "Central limit theorems for point processes," Stochastic Processes and their Applications, Elsevier, vol. 12(2), pages 171-186, March.
    5. Peter Diggle & Pingping Zheng & Peter Durr, 2005. "Nonparametric estimation of spatial segregation in a multivariate point process: bovine tuberculosis in Cornwall, UK," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 54(3), pages 645-658, June.
    6. Frederic Paik Schoenberg, 2004. "Testing Separability in Spatial-Temporal Marked Point Processes," Biometrics, The International Biometric Society, vol. 60(2), pages 471-481, June.
    7. Roger D. Peng & Frederic Paik Schoenberg & James A. Woods, 2005. "A Space-Time Conditional Intensity Model for Evaluating a Wildfire Hazard Index," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 26-35, March.
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    Cited by:

    1. Zhang, Tonglin & Mateu, Jorge, 2019. "Substationarity for spatial point processes," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 22-36.

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