This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# MaxStableDistribution

 MaxStableDistribution represents a generalized maximum extreme value distribution with location parameter , scale parameter , and shape parameter .
• The generalized maximum extreme value distribution gives the asymptotic distribution of the maximum value in a sample from a distribution such as the normal, Cauchy, or beta distribution.
• The probability density for value in a generalized maximum extreme value distribution is proportional to for and zero otherwise.
Probability density function:
Cumulative distribution function:
Mean and variance:
Median:
Probability density function:
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Cumulative distribution function:
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Mean and variance:
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Median:
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 Scope   (7)
Generate a set of pseudorandom numbers that follow generalized maximum extreme value distribution:
Compare its histogram to the PDF:
Distribution parameters estimation:
Estimate the distribution parameters from sample data:
Compare a density histogram of the sample with the PDF of the estimated distribution:
Skewness depends only on the shape parameter:
Limiting values:
Find the shape parameter for which the distribution is symmetric:
Skewness has opposite sign to skewness of MinStableDistribution:
Kurtosis depends only on the shape parameter:
Limiting values:
Kurtosis attains its minimum:
Kurtosis is the same as kurtosis of MinStableDistribution:
Different moments with closed forms as functions of parameters:
Hazard function:
Quantile function:
 Applications   (1)
MaxStableDistribution can be used to model fiber strength. Consider the tensile strength of Indian cotton given in grams:
Fit the distribution into the data:
Compare the histogram of the data with the PDF of the estimated distribution:
Find the average tensile strength:
Find the probability that the fiber strength is at least 10:
Simulate the fiber strength for the next 50 samples:
Parameter influence on the CDF for each :
MaxStableDistribution is closed under translation and scaling by a positive factor:
CDF of MaxStableDistribution solves the stability postulate equation:
Verify solution for :
Find the limit of :
Relationships to other distributions:
ExtremeValueDistribution is a special case of a generalized maximum extreme value distribution:
FrechetDistribution is a special case of a generalized maximum extreme value distribution:
Generalized maximum extreme value distribution is related to WeibullDistribution:
Generalized maximum extreme value distribution is related to GumbelDistribution:
Generalized maximum extreme value distribution is related to MinStableDistribution:
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