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Encyclopedia > 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. Extreme value theory is important for assessing risk for highly unusual events, such as 100-year floods. Statistics is a type of data analysis whose practice includes the planning, summarizing, and interpreting of observations of a system possibly followed by predicting or forecasting of future events based on a mathematical model of the system being observed. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... Risk is the potential harm that may arise from some present process or from some future event. ... A one-hundred year flood is calculated to be the maximum level of flood water to be expected in an average one-hundred-year period. ...


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 distributions are the limiting distributions for the minimum or the maximum of a very large collection of random observations from the same arbitrary distribution. 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

Contents


Applications

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

Look up Flood in Wiktionary, the free dictionary A flood (in Old English flod, a word common to Teutonic languages; compare German Flut, Dutch vloed from the same root as is seen in flow, float) is an overflow of water, an expanse of water submerging land, a deluge. ... Insurance, in law and economics, is a form of risk management primarily used to hedge against the risk of potential financial loss. ... Equity Risk is the risk that ones investments will depreciate due to stock market dynamics causing one to lose money. ... Market risk is the risk that the value of your investment will decrease due to moves in market factors. ... Freak waves, also known as rogue waves or monster waves, are relatively large and spontaneous ocean surface waves which can sink even medium-large ships. ...

History of extreme value theory

The field of extreme value theory was founded by the German mathematician, pacifist, and anti-Nazi campaigner Emil Julius Gumbel who described the Gumbel distribution in the 1950s. Emil Julius Gumbel (July 18, 1891 - September 10, 1966), German mathematician, pacifist and anti-Nazi campaigner. ... This article needs cleanup. ... // Events and trends The 1950s in Western society was marked with a sharp rise in the economy for the first time in almost 30 years and return to the 1920s-type consumer society built on credit and boom-times, as well as the height of the baby-boom from returning...


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 (hydro- meaning water, and -geology meaning the study of rocks) is the part of hydrology that deals with the distribution and movement of groundwater in the soil and rocks of the Earths crust (commonly in aquifers). ... Cumulus clouds This article needs to be cleaned up to conform to a higher standard of quality. ... It is requested that this article, or a section of this article, be expanded. ... Freak waves, also known as rogue waves or monster waves, are relatively large and spontaneous ocean surface waves which can sink even medium-large ships. ... Probability The Doctrine of Chances Author: Abraham de Moivre Publication data: 1738 (2nd ed. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ...

External links


  Results from FactBites:
 
Extreme value theory - Wikipedia, the free encyclopedia (399 words)
Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions.
Extreme value theory is important for assessing risk for highly unusual events, such as 100-year floods.
Extreme value distributions are the limiting distributions for the minimum or the maximum of a very large collection of random observations from the same arbitrary distribution.
  More results at FactBites »

 
 

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