IDEAS home Printed from
   My bibliography  Save this article

The foundations of statistics with black swans


  • Chichilnisky, Graciela


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

    Download full text from publisher

    File URL:
    Download Restriction: Full text for ScienceDirect subscribers only

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    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. Chichilnisky, Graciela, 2009. "The Limits Of Econometrics: Nonparametric Estimation In Hilbert Spaces," Econometric Theory, Cambridge University Press, vol. 25(04), pages 1070-1086, August.
    8. Chichilnisky, Graciela, 2009. "The topology of fear," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 807-816, December.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    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.


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matsoc:v:59:y:2010:i:2:p:184-192. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Dana Niculescu). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.