Extreme value theory

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



Two approaches exist today:

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, called Peak Over Threshold models (POT). 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 used for cases where Theorem I applies, which creates problems with the basic model assumptions.

Extreme value distributions are the limiting distributions for the minimum or the maximum of a very large collection of independent random variables from the same arbitrary distribution. Emil Julius Gumbel (1958) showed that for any well-behaved initial distribution (i.e., F(x) is continuous and has an inverse), only a few models are needed, depending on whether you are interested in the maximum or the minimum, and also if the observations are bounded above or below.


Applications of extreme value theory include predicting the probability distribution of:

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