# Extreme Value

• MaximumExtreme (Mode, Scale)

• MinimumExtreme (Mode, Scale)

Extreme Value equations

Crystal Ball parameter restrictions

Extreme Value distribution is more commonly known as the Gumbel distribution. Extreme value distributions model the maximum, or minimum, of a set of random variables that all follow the same distribution. Thus, the appropriate extreme value distribution depends on the underlying distribution of the random variable.

The Extreme Value and Maximum Extreme Value distribution models the maximum, and the Minimum Extreme Value distribution the minimum of a set of random variables that have an underlying distribution belonging to the Exponential family, e.g. Exponential, Gamma, Weibull, Normal, Lognormal, Logistic and itself.

Examples of the ExtremeValue distribution (Maximum Extreme value) and the Minimum Extreme value distribution are given below:

#### More on Extreme Value Distributions

The Gumbel distribution is one (the most useful one) of three extreme value distributions. All three distributions take the form   $//$where   $//$is defined as e-x when g = 0 which is the standard (a = 0, b = 1) Gumbel distribution. The other two distributions are a version of the Weibull distribution (g < 0) and the Frechet distribution (g > 0) though the latter is not popularly used.

The theory of extreme values says that the largest or smallest value from a set of values drawn from the same parent distribution tends to an asymptotic distribution that only depends on the tail of the parent distribution. The Frechet distribution is the extreme value distribution for parent distributions of the form of Pareto, Student-t, Cauchy, log-Gamma and itself. The Weibull distribution is the extreme value distribution for Beta, Uniform and Weibull distributed variables but the convergence can be very slow.

#### Uses

Engineers are often interested in extreme values of a parameter (like minimum strength, maximum impinging force) because they are the values that determine whether a system will potentially fail. For example: wind strengths impinging on a building - it must be designed to sustain the largest wind with minimum damage within the bounds of the finances available to build it; maximum wave height for designing offshore platforms, breakwaters and dikes; pollution emissions for a factory to ensure that, at its maximum, it will fall below the legal limit; determining the strength of a chain, since it is equal to the strength of its weakest link; modeling the extremes of meteorological events since these cause the greatest impact.

The form of the Gumbel distribution provided by the ExtremeValue distribution is to model the maximum extreme. In the case that the minimum extreme value distribution is not available in the simulation software package, for a variable that has an exponential family lower tail, the minimum extreme value distribution can be modeled by reversing the sign of the data (X). So, if you have a set of minimum data, change the sign of each data point, fit the above Extreme Value distribution to give a ExtremeValue(m,s), and then reverse the sign again, i.e. Lower extreme = - ExtremeValue(m,s). How to construct a minimum extreme value distribution is shown in the model  Fitting_ExtValue.

The parameters of the Extreme Value distribution are usually determined by data fitting. As a guide, if there are n data points available for a parameter, the lowest (or highest) √n points can be used to attempt a fit. Gumbel (1958) provides an old but still excellent treatise on extreme value theory.

For the Gumbel distribution, a  is a location parameter and b is a scale parameter, x always appears in the form (x-a)/b. This means that all Gumbel distributions have the same shape, with skewness = 1.139548 and kurtosis = 5.4.

The Gumbel distribution is also sometimes called the LogWeibull, Gompertz or Fisher-Tippett distribution.

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