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Maximization of statistical heterogeneity: From Shannon’s entropy to Gini’s index

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  • Eliazar, Iddo
  • Sokolov, Igor M.

Abstract

Different fields of Science apply different quantitative gauges to measure statistical heterogeneity. Statistical Physics and Information Theory commonly use Shannon’s entropy which measures the randomness of probability laws, whereas Economics and the Social Sciences commonly use Gini’s index which measures the evenness of probability laws. Motivated by the principle of maximal entropy, we explore the maximization of statistical heterogeneity–for probability laws with a given mean–in the four following scenarios: (i) Shannon entropy maximization subject to a given dispersion level; (ii) Gini index maximization subject to a given dispersion level; (iii) Shannon entropy maximization subject to a given Gini index; (iv) Gini index maximization subject to a given Shannon entropy. Analysis of these four scenarios results in four different classes of heterogeneity-maximizing probability laws–yielding an in-depth description of both the marked differences and the interplay between the Physical “randomness-based” and the Economic “evenness-based” approaches to the maximization of statistical heterogeneity.

Suggested Citation

  • Eliazar, Iddo & Sokolov, Igor M., 2010. "Maximization of statistical heterogeneity: From Shannon’s entropy to Gini’s index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(16), pages 3023-3038.
    • Handle: RePEc:eee:phsmap:v:389:y:2010:i:16:p:3023-3038
    DOI: 10.1016/j.physa.2010.03.045
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    References listed on IDEAS

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    1. Alfarano, Simone & Milakovic, Mishael, 2008. "Does classical competition explain the statistical features of firm growth?," Economics Letters, Elsevier, vol. 101(3), pages 272-274, December.
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    Cited by:

    1. Viktor Stojkoski & Petar Jolakoski & Arnab Pal & Trifce Sandev & Ljupco Kocarev & Ralf Metzler, 2021. "Income inequality and mobility in geometric Brownian motion with stochastic resetting: theoretical results and empirical evidence of non-ergodicity," Papers 2109.01822, arXiv.org.
    2. Khosravi Tanak, A. & Mohtashami Borzadaran, G.R. & Ahmadi, J., 2015. "Entropy maximization under the constraints on the generalized Gini index and its application in modeling income distributions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 438(C), pages 657-666.
    3. Khosravi Tanak, A. & Mohtashami Borzadaran, G.R. & Ahmadi, J., 2017. "Maximum Tsallis entropy with generalized Gini and Gini mean difference indices constraints," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 554-560.
    4. Khosravi Tanak, A. & Mohtashami Borzadaran, G.R. & Ahmadi, Jafar, 2018. "New functional forms of Lorenz curves by maximizing Tsallis entropy of income share function under the constraint on generalized Gini index," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 511(C), pages 280-288.
    5. Fontanari, Andrea & Taleb, Nassim Nicholas & Cirillo, Pasquale, 2018. "Gini estimation under infinite variance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 502(C), pages 256-269.

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