Maximum Extreme Value distribution | Vose Software

# Maximum Extreme Value distribution

Format: ExtValueMax(a, b)

The Maximum Extreme Value distribution models the maximum 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.

## Uses

We are often interested in extreme values of a parameter (like minimum strength, maximum impinging force, maximum change in a stock price) 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 ExtremeValueMax distribution in ModelRisk models a Gumbel distribution for the maximum extreme. The minimum extreme distribution, for a variable that has an exponential family lower tail, is given by the complementary ExtremeValueMin distribution.

Note that the Extreme Value distributions are asymptotic results, meaning that the probability distribution of the maximum of a set of n independent values drawn from some distribution approaches the Extreme Value distributions only as n approaches infinity. For practical purposes it will usually be more accurate, practical and transparent to use one of the following ModelRisk functions:

VoseLargest

VoseLargestSet

VoseKthLargest

VoseSmallest

VoseSmallestSet

VoseKthSmallest

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. Embrechts et al (1999) provides a more up-to-date, though rather academic, study of extremes.

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 (max), -1.139548 (min) and kurtosis = 5.4.

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

## ModelRisk functions added to Microsoft Excel for the Maximum Extreme Value distribution

VoseExtValueMax generates random values from this distribution for Monte Carlo simulation, or calculates a percentile if used with a U parameter.

VoseExtValueMaxObject constructs a distribution object for this distribution.

VoseExtValueMaxProb returns the probability density or cumulative distribution function for this distribution.

VoseExtValueMaxProb10 returns the log10 of the probability density or cumulative distribution function.

VoseExtValueMaxFit generates values from this distribution fitted to data, or calculates a percentile from the fitted distribution.

VoseExtValueMaxFitObject constructs a distribution object of this distribution fitted to data.

VoseExtValueMaxFitP returns the parameters of this distribution fitted to data.