The foundations of statistics with black swans
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  and ; 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.
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- Chichilnisky, Graciela, 2009. "The Limits Of Econometrics: Nonparametric Estimation In Hilbert Spaces," Econometric Theory, Cambridge University Press, vol. 25(04), pages 1070-1086, August.
- F J Anscombe & R J Aumann, 2000. "A Definition of Subjective Probability," Levine's Working Paper Archive 7591, David K. Levine.
- 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.
- 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.
- 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.
- 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.
- Chichilnisky, G. & Heal, G.M., 1995. "Social Choice with Infinite Populations: Construction of a Rule and Impossibility Results," Papers 95-19, Columbia - Graduate School of Business.
- Chichilnisky, Graciela, 2009. "The topology of fear," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 807-816, December.
- 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.
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