# Optimal Statistical Decisions with Catastrophic Risks

*Air Force Office of Scientific Research, FA-9550-09-1-0467*

Principal Investigator: Graciela Chichilnisky

__Abstract__

The proposal is to extend classical statistical analysis - including Bayesian statistical inference - to an axiomatic treatment for samples that include catastrophic events. The strategy for the research is as follows: the theory of choice under uncertainty is distinct from but parallels the foundations of statistical analysis. The axioms for choice under uncertainty describe how we rank our choices, what we prefer, while relative likelihood ranks the frequency of what is observed in reality. In both cases, there are axioms to explain the primitive concepts “is more likely than” for probability, and “is preferred to” for utility. Following this line of argument the new research proposed here will extend earlier axioms and results by the author on choice under uncertainty to a completely new area of statistical inference and analysis, in order to explain how we observe and predict in samples that include catastrophic events. We expect the research will provide an axiomatic characterization of probability distributions that explain the persistent observation of power laws, “heavy tailed” distributions in financial models and “black swans” in the natural hazards literature on earthquakes and floods.

The results should achieve the following goals: (i) show how standard statistical analysis based on existing relative likelihood axioms underestimate the incidence of catastrophes, and thus unnecessarily increase losses after the fact; (ii) develop new relative likelihood axioms that correct this bias; (iii) obtain a representation theorem that characterizes the likelihoods or probability distributions that the new axioms imply; (iv) introduce new algorithms and study numerical approximations of the new types of decision rules when catastrophic events are considered; and (v) practical applications to various branches of decision theory including Bayesian analysis, non-parametric econometrics, sustainable development issues arising from climate change and biodiversity extinction, as well as decisions involving national security issues.

__Working Papers and Publications__

Chichilnisky, Graciela, 2010. "The foundations of statistics with black swans," Mathematical Social Sciences, Elsevier, vol. 59(2), pages 184-192, March.

Full text: http://www.chichilnisky.com/pdfs/black-swans.pdf

Graciela Chichilnisky, 2010. “The Foundations of Probability with Black Swans,” Journal of Probability and Statistics, vol. 2010, Article ID 838240, 11 pages,

Full text: http://www.hindawi.com/journals/jps/2010/838240.html

Chichilnisky, Graciela, 2009. "The topology of fear," Journal of Mathematical Economics, Elsevier, vol. 45(12), pages 807-816, December.

Full text: http://www.chichilnisky.com/pdfs/Topology%20of%20Fear.pdf

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.

Full text: http://www.springerlink.com/content/uu37208q7g8052k8/

Graciela Chichilnisky & Peter Eisenberger, 2009. "Asteroids: Assessing Catastrophic Risks," Working Papers 09-13, LAMETA, Universtiy of Montpellier, revised Nov 2009.

Full text: http://www.chichilnisky.com/pdfs/Asteroids_JPS.pdf

Graciela Chichilnisky, 2009. “Catastrophic Risks” International Journal of Green Economics

Full text: http://www.chichilnisky.com/pdfs/catastrophic-risks.pdf