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Fitting power-laws in empirical data with estimators that work for all exponents

Author

Listed:
  • Rudolf Hanel
  • Bernat Corominas-Murtra
  • Bo Liu
  • Stefan Thurner

Abstract

Most standard methods based on maximum likelihood (ML) estimates of power-law exponents can only be reliably used to identify exponents smaller than minus one. The argument that power laws are otherwise not normalizable, depends on the underlying sample space the data is drawn from, and is true only for sample spaces that are unbounded from above. Power-laws obtained from bounded sample spaces (as is the case for practically all data related problems) are always free of such limitations and maximum likelihood estimates can be obtained for arbitrary powers without restrictions. Here we first derive the appropriate ML estimator for arbitrary exponents of power-law distributions on bounded discrete sample spaces. We then show that an almost identical estimator also works perfectly for continuous data. We implemented this ML estimator and discuss its performance with previous attempts. We present a general recipe of how to use these estimators and present the associated computer codes.

Suggested Citation

  • Rudolf Hanel & Bernat Corominas-Murtra & Bo Liu & Stefan Thurner, 2017. "Fitting power-laws in empirical data with estimators that work for all exponents," PLOS ONE, Public Library of Science, vol. 12(2), pages 1-15, February.
  • Handle: RePEc:plo:pone00:0170920
    DOI: 10.1371/journal.pone.0170920
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    Cited by:

    1. Wim Ectors & Bruno Kochan & Davy Janssens & Tom Bellemans & Geert Wets, 2019. "Exploratory analysis of Zipf’s universal power law in activity schedules," Transportation, Springer, vol. 46(5), pages 1689-1712, October.
    2. Nelson, Kenric P., 2022. "Independent Approximates enable closed-form estimation of heavy-tailed distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 601(C).
    3. Wei Zhu & Ding Ma & Zhigang Zhao & Renzhong Guo, 2020. "Investigating the Complexity of Spatial Interactions between Different Administrative Units in China Using Flickr Data," Sustainability, MDPI, vol. 12(22), pages 1-12, November.
    4. Aguilar-Velázquez, D. & Guzmán-Vargas, L., 2017. "Synchronization and 1/f signals in interacting small-world networks," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 418-425.
    5. Nasrolahzadeh, Mahda & Mohammadpoory, Zeynab & Haddadnia, Javad, 2023. "Indices from visibility graph complexity of spontaneous speech signal: An efficient nonlinear tool for Alzheimer's disease diagnosis," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).

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