BUILT-IN MATHEMATICA SYMBOL

MaxStableDistribution

represents a generalized maximum extreme value distribution with location parameter

, scale parameter

, and shape parameter

.

MORE INFORMATION

MaxStableDistribution is also known as Fisher-Tippet distribution.

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.

MaxStableDistribution allows and to be any real numbers and to be any positive real number.

MaxStableDistribution can be used with such functions as Mean, CDF, and RandomVariate.

EXAMPLES

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Basic Examples (4)

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:

Properties & Relations (9)

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: