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In probability theory and statistics, the generalized extreme value (GEV) distribution [2] is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution ...
Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering , finance , economics , earth sciences , traffic prediction, and geological engineering .
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
The standard Gumbel distribution is the case where = and = with cumulative distribution function = ()and probability density function = (+).In this case the mode is 0, the median is ( ()), the mean is (the Euler–Mascheroni constant), and the standard deviation is /
The Fréchet distribution, also known as inverse Weibull distribution, [2] [3] is a special case of the generalized extreme value distribution. It has the cumulative distribution function ( ) = > . where α > 0 is a shape parameter.
The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution , above. The study of conditions for convergence of to particular cases of the generalized extreme value distribution began with Mises (1936) [3] [5] [4] and was further developed by Gnedenko (1943).
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In null-hypothesis significance testing, the p-value [note 1] is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. [2] [3] A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis.