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Testing the Pareto against the lognormal distributions with the uniformly most powerful unbiased test applied to the distribution of cities

Yannick Malevergne1,2,3, Vladilen Pisarenko4, and Didier Sornette3,5

  • 1Université de Lyon-Université de Saint-Etienne, Coactis E.A. 4161, 42023 Saint-Etienne, France
  • 2EMLYON Business School, Cefra, 69134 Ecully, France
  • 3Department of Management, Technology, and Economics, ETH Zurich, Switzerland
  • 4International Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Science, Moscow, Russia
  • 5Swiss Finance Institute, c/o University of Geneva, 40 blvd. Du Pont d’Arve CH-1211 Geneva 4, Switzerland

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Phys. Rev. E 83, 036111 – Published 22 March, 2011

DOI: https://doi.org/10.1103/PhysRevE.83.036111

Abstract

Fat-tail distributions of sizes abound in natural, physical, economic, and social systems. The lognormal and the power laws have historically competed for recognition with sometimes closely related generating processes and hard-to-distinguish tail properties. This state-of-affair is illustrated with the debate between Eeckhout [Amer. Econ. Rev. 94, 1429 (2004)] and Levy [Amer. Econ. Rev. 99, 1672 (2009)] on the validity of Zipf’s law for US city sizes. By using a uniformly most powerful unbiased (UMPU) test between the lognormal and the power-laws, we show that conclusive results can be achieved to end this debate. We advocate the UMPU test as a systematic tool to address similar controversies in the literature of many disciplines involving power laws, scaling, “fat” or “heavy” tails. In order to demonstrate that our procedure works for data sets other than the US city size distribution, we also briefly present the results obtained for the power-law tail of the distribution of personal identity (ID) losses, which constitute one of the major emergent risks at the interface between cyberspace and reality.

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Phys. Rev. E 83, 036111– Published 22 March, 2011
  • Received 9 September 2010

©2011 American Physical Society

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