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A KPSS Test for Stationarity for Spatial Point Processes

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  • Yongtao Guan

Abstract

Summary We propose a formal method to test stationarity for spatial point processes. The proposed test statistic is based on the integrated squared deviations of observed counts of events from their means estimated under stationarity. We show that the resulting test statistic converges in distribution to a functional of a two‐dimensional Brownian motion. To conduct the test, we compare the calculated statistic with the upper tail critical values of this functional. Our method requires only a weak dependence condition on the process but does not assume any parametric model for it. As a result, it can be applied to a wide class of spatial point process models. We study the efficacy of the test through both simulations and applications to two real data examples that were previously suspected to be nonstationary based on graphical evidence. Our test formally confirmed the suspected nonstationarity for both data.

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  • Yongtao Guan, 2008. "A KPSS Test for Stationarity for Spatial Point Processes," Biometrics, The International Biometric Society, vol. 64(3), pages 800-806, September.
    • Handle: RePEc:bla:biomet:v:64:y:2008:i:3:p:800-806
    DOI: 10.1111/j.1541-0420.2007.00977.x
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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. Dimitris N. Politis & Michael Sherman, 2001. "Moment estimation for statistics from marked point processes," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 63(2), pages 261-275.
    3. A. J. Baddeley & J. Møller & R. Waagepetersen, 2000. "Non‐ and semi‐parametric estimation of interaction in inhomogeneous point patterns," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 54(3), pages 329-350, November.
    4. Giraitis, Liudas & Leipus, Remigijus & Philippe, Anne, 2006. "A Test For Stationarity Versus Trends And Unit Roots For A Wide Class Of Dependent Errors," Econometric Theory, Cambridge University Press, vol. 22(6), pages 989-1029, December.
    5. Bart Hobijn & Philip Hans Franses & Marius Ooms, 2004. "Generalizations of the KPSS‐test for stationarity," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 58(4), pages 483-502, November.
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    Cited by:

    1. 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.
    2. Sung Nok Chiu & Kwong Ip Liu, 2013. "Stationarity Tests for Spatial Point Processes using Discrepancies," Biometrics, The International Biometric Society, vol. 69(2), pages 497-507, June.
    3. Yehua Li & Yongtao Guan, 2014. "Functional Principal Component Analysis of Spatiotemporal Point Processes With Applications in Disease Surveillance," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 109(507), pages 1205-1215, September.
    4. Heinrich Lothar & Klein Stella, 2011. "Central limit theorem for the integrated squared error of the empirical second-order product density and goodness-of-fit tests for stationary point processes," Statistics & Risk Modeling, De Gruyter, vol. 28(4), pages 359-387, December.
    5. Zhang, Tonglin & Mateu, Jorge, 2019. "Substationarity for spatial point processes," Journal of Multivariate Analysis, Elsevier, vol. 171(C), pages 22-36.
    6. Xinyu Zhou & Wei Wu, 2024. "Statistical Depth in Spatial Point Process," Mathematics, MDPI, vol. 12(4), pages 1-20, February.
    7. Ka Yiu Wong & Dietrich Stoyan, 2021. "Poles of pair correlation functions: When they are real?," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 73(2), pages 425-440, April.

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