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The foundations of statistics with black swans

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  • Chichilnisky, Graciela

Abstract

We extend the foundation of statistics to integrate rare events that are potentially catastrophic, called black swans.These include natural hazards, regime change in complex systems, market crashes, catastrophic climate change and major episodes of species extinction. Classic statistics and physics treat such events as 'outliers' and often disregard them. We propose a new axiomatization of subjective probability requiring equal treatment for rare and frequent events, and characterize the likelihoods or subjective probabilities that the axioms imply. These coincide with countably additive measures and yield normal distributions when the sample has no black swans. When the sample includes black swans, the new likelihoods are represented by a combination of countable and finitely additive measures with both parts present. The axioms were introduced in [5] and [6]; they extend the axiomatic foundations of Savage (1954), Villegas (1964) and Arrow (1971) and they are valid for bounded and unbounded samples (Chichilnisky, 1996b). The finitely additive measures assign more weight to rare events than do standard distributions and in that sense explain the persistent observation of power laws and 'heavy tails' that eludes classic theory.

Suggested Citation

  • Chichilnisky, Graciela, 2010. "The foundations of statistics with black swans," Mathematical Social Sciences, Elsevier, vol. 59(2), pages 184-192, March.
  • Handle: RePEc:eee:matsoc:v:59:y:2010:i:2:p:184-192
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    References listed on IDEAS

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    1. Chichilnisky, Graciela, 2000. "An axiomatic approach to choice under uncertainty with catastrophic risks," Resource and Energy Economics, Elsevier, vol. 22(3), pages 221-231, July.
    2. F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
    3. Chichilnisky, Graciela & Wu, Ho-Mou, 2006. "General equilibrium with endogenous uncertainty and default," Journal of Mathematical Economics, Elsevier, vol. 42(4-5), pages 499-524, August.
    4. Olivier Chanel & Graciela Chichilnisky, 2009. "The influence of fear in decisions: Experimental evidence," Journal of Risk and Uncertainty, Springer, vol. 39(3), pages 271-298, December.
    5. Graciela Chichilnisky, 1996. "An axiomatic approach to sustainable development," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 13(2), pages 231-257, April.
    6. Graciela Chichilnisky & Geoffrey Heal, 1997. "Social choice with infinite populations: construction of a rule and impossibility results," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 14(2), pages 303-318.
    7. Graciela Chichilnisky, 2011. "The Limits of Econometrics: Nonparametric Estimation in Hilbert Spaces," Chapters, in: Miroslav Verbic (ed.), Advances in Econometrics - Theory and Applications, IntechOpen.
    8. Chichilnisky, Graciela, 2009. "The topology of fear," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 807-816, December.
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    Cited by:

    1. Graciela Chichilnisky, 2012. "Sustainable markets with short sales," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 49(2), pages 293-307, February.
    2. Norman Schofield, 2015. "Climate Change, Collapse and Social Choice Theory," Czech Economic Review, Charles University Prague, Faculty of Social Sciences, Institute of Economic Studies, vol. 9(1), pages 007-035, October.
    3. Kitti, Mitri, 2018. "Sustainable social choice under risk," Mathematical Social Sciences, Elsevier, vol. 94(C), pages 19-31.
    4. M. Ali Khan & Metin Uyanik, 2020. "Binary Relations in Mathematical Economics: On the Continuity, Additivity and Monotonicity Postulates in Eilenberg, Villegas and DeGroot," Papers 2007.01952, arXiv.org.
    5. Harald de Bruijn & Andreas Größler & Nuno Videira, 2020. "Antifragility as a design criterion for modelling dynamic systems," Systems Research and Behavioral Science, Wiley Blackwell, vol. 37(1), pages 23-37, January.

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