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ExtremeValueDistribution

BUILT-IN WOLFRAM LANGUAGE SYMBOL

# ExtremeValueDistribution

ExtremeValueDistribution[α,β]

represents an extreme value distribution with location parameter α and scale parameter β.

## DetailsDetails

- The extreme value distribution gives the asymptotic distribution of the maximum value in a sample from a distribution such as the normal distribution.
- The probability density for value in an extreme value distribution is proportional to . »
- The asymptotic distribution of the minimum value, also sometimes called an extreme value distribution, is implemented in the Wolfram Language as GumbelDistribution. »
- ExtremeValueDistribution allows α to be any real number and β to be any positive real number.
- ExtremeValueDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

## BackgroundBackground

- ExtremeValueDistribution[α,β] represents a continuous statistical distribution defined over the set of real numbers and parametrized by a real number α, called a "location parameter", and a positive real number β, called a "scale parameter". While the overall behavior of the probability distribution function (PDF) of the extreme value distribution is smooth and unimodal, the parameters α and β determine the horizontal location and overall height and steepness, respectively, of the PDF. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The extreme value distribution may also be referred to as the type-1 extreme value distribution, the doubly exponential distribution (not to be confused with LaplaceDistribution, which may also be called doubly exponential), the log-Weibull distribution (WeibullDistribution), or the Gumbel-type distribution (not to be confused with GumbelDistribution).
- While ExtremeValueDistribution is sometimes referred to as "the" extreme value distribution, it is actually one of four distributions (along with GumbelDistribution, FrechetDistribution, and WeibullDistribution) classified under the general heading "extreme value distribution". The extreme value distribution is obtained as the limiting distribution of greatest values in random samples of increasing size, and because its PDF is doubly exponential (i.e. is of the form Exp[-Exp[…]]), the graph of the distribution has more exaggerated features (like higher peaks and thinner tails), a property unique among distributions. It was introduced in the early 1920s to address the need in astronomy to utilize or reject outlying observations. A cornerstone in the field known as extreme value theory, the extreme value distribution is widely utilized to describe situations which are "extremely unlikely" (i.e. those in which datasets consist of variates with extreme deviations from the median), e.g. extreme floods, catastrophic insurance losses, and large wildfires. Moreover, the extreme value distribution can be used in biology as a tool to model evolutionary genetic mutations.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an extreme value distribution. Distributed[x,ExtremeValueDistribution[α,β]], written more concisely as , can be used to assert that a random variable x is distributed according to an extreme value distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
- The probability distribution and cumulative density functions may be given using PDF[ExtremeValueDistribution[α,β],x] and CDF[ExtremeValueDistribution[α,β],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
- DistributionFitTest can be used to test if a given dataset is consistent with an extreme value distribution, EstimatedDistribution to estimate an extreme value parametric distribution from given data, and FindDistributionParameters to fit data to an extreme value distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic extreme value distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic extreme value distribution.
- TransformedDistribution can be used to represent a transformed extreme value distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an extreme value distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving extreme value distributions.
- The extreme value distribution is related to a number of other distributions. As mentioned previously, ExtremeValueDistribution is one of a number of distributions, including GumbelDistribution, FrechetDistribution, and WeibullDistribution, that are used to describe extremely unlikely events. This qualitative relationship also has quantitative components, as the PDF of ExtremeValueDistribution[α,β] is precisely the same as that of TransformedDistribution[-u,u GumbelDistribution[-α,β]] and TransformedDistribution[-β α Log[u/β]+α,u WeibullDistribution[α,β]]. ExtremeValueDistribution is a transformation of ExponentialDistribution and MinStableDistribution, and is a special case of MaxStableDistribution, in the sense that the PDF of ExtremeValueDistribution[α,β] is identical to that of MaxStableDistribution[α,β,0]. ExtremeValueDistribution is also related to ExpGammaDistribution, ExponentialDistribution, and LogisticDistribution.

Introduced in 2007

(6.0)

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