Examining Tail Distributions of Moran’s I Statistic through Intensive Simulations

  • Ikuho YamadaEmail author
  • Atsuyuki Okabe
Part of the Advances in Geographic Information Science book series (AGIS)


Moran’s I statistic is arguably the most widely applied method for testing spatial autocorrelation in areal datasets. This study examines its probability distribution through intensive simulation experiments with a specific focus on its tails, which are the most important for statistical testing. While Moran’s I statistic is known to be asymptotically normal when the number of zones in a study region is sufficiently large, results of this study indicate that the normality is hardly achievable even when the number of zones is increased up to 2,500. Our results also suggest that discrepancies from the normality are more serious for target variables with larger skewness and kurtosis. Simulation-based testing, which does not rely upon the assumed normality of I, is thus recommended. This study proposes to carry out multiple sets of simulations and use the mean of simulated percentage point values as critical values to reduce instability inherent in simulation-based approaches.


Moran’s I statistic Spatial autocorrelation Statistical testing Spatial statistics Probability distributions 



This work was supported by JSPS KAKENHI Grant Number 24650606. The authors greatly appreciate helpful comments and information from anonymous reviewers as well as attendees of the Joint International Conference on Geospatial Theory, Processing, Modelling and Applications in Toronto in 2014.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Chuo UniversityBunkyo-ku, TokyoJapan
  2. 2.Aoyama Gakuin UniversityShibuya-ku, TokyoJapan

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