The Extreme value theory reference article from the English Wikipedia on 24-Apr-2004 (provided by Fixed Reference: snapshots of Wikipedia from wikipedia.org)

Extreme value theory

Extreme value theory is a branch of statistics dealing with the extreme deviations from the mean of probability distributions. The general theory sets out to assess the type of probability distributions generated by processes.

Two approaches exist today:

1. most common at this moment is the tail fitting approach based on the second theorem in extreme value theory (Theorem II Pickands(1975), Balkema and de Haan(1974)).
2. Basic theory approach as described in the book by Burry (reference 2).

In general this conforms to the first theorem in extreme value theory (Theorem I Fisher and Tippett(1928), and Gnedenko(1943)).

The difference between the two theorems is due to the nature of the data generation. For theorem I the data are generated in full range, while in theorem II data is only generated when it surpasses a certain threshold (POT's models or Peak Over Threshold). The POT approach as been developped largely in the insurance business, where only losses (pay outs) above a certain threshold are accessible to the insurance company. Strangely this approach is often applied to theorem I cases which poses problems with the basic model assumptions.

Extreme value theory is important for assessing risk for highly unusual events, such as 100-year floods.

Applications of extreme value theory:

• predicting extreme floods
• predicting the amounts of large insurance losses
• predicting equity risks
• predicting day to day market risk

History of extreme value theory

Founded by the German mathematician, pacifist, and anti-Nazi campaigner Emil Julius Gumbel who described the Gumbel distribution in the 1950s.

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References:

• Gumbel, E.J.(1958). Statistics of Extremes. Columbia University Press.
• Burry K.V. (1975). Statistical methods in applied science. John Wiley & Sons.

See also:

• hydrogeology, meteorology, extreme weather

External links

• Easy non-mathematical introduction
• Extreme value theory group at Chalmers University
• The Extreme Value Approach to VaR ? An Introduction
• Extreme Value Theory for Tail-Related Risk Measures
• Extreme value theory an empirical analysis of equity risk
• http://www.itl.nist.gov/div898/handbook/apr/section1/apr163.htm

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