Extreme value theory

Extreme value theory is a branch of statistics dealing with the extreme deviations from the median 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 has been developed 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 include predicting the probability distribution of:

• extreme floods
• the amounts of large insurance losses
• equity risks
• day to day market risk
• the size of freak waves

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.

References

• Gumbel, E.J. (1958). Statistics of Extremes. Columbia University Press.
• Burry K.V. (1975). Statistical Methods in Applied Science. John Wiley & Sons.
• Pickands, J. (1975). Statistical inference using extreme order statistics, Annals of Statistics, 3, 119-131.
• Balkema, A., and L. de Haan (1974). Residual life time at great age, Annals of Probability, 2, 792-804.
• Fisher, R.A., and L. H. C. Tippett (1928). Limiting forms of the frequency distribution of the largest and smallest member of a sample, Proc. Cambridge Phil. Soc., 24, 180-190.
• Gnedenko, B.V. (1943), Sur la distribution limite du terme maximum d'une serie aleatoire, Annals of Mathematics, 44, 423-453

See also

• hydrogeology, meteorology, extreme weather, freak wave
• Important publications in extreme value theory