BUILT-IN MATHEMATICA SYMBOL

MinStableDistribution

represents a generalized minimum extreme value distribution with location parameter

, scale parameter

, and shape parameter

.

MORE INFORMATION

MinStableDistribution is also known as Fisher-Tippet distribution.

The generalized minimum extreme value distribution gives the asymptotic distribution of the minimum value in a sample from a distribution such as the normal, Cauchy, or beta distribution.

The probability density for value in a generalized minimum extreme value distribution is proportional to for and zero otherwise.

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

MinStableDistribution 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 minimum 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:

The distribution is symmetric for:

Skewness has opposite sign to skewness of MaxStableDistribution:

Kurtosis depends only on the shape parameter:

Limiting values:

Kurtosis attains its minimum:

Kurtosis is the same as kurtosis of MaxStableDistribution:

Different moments with closed forms as functions of parameters:

Hazard function:

Quantile function:

Applications (4)

MinStableDistribution can be used to model the annual minimum mean daily flows. Consider the Mahanadi river and the minimum flows given in cubic meters per second:

Fit MinStableDistribution to the data:

Compare the histogram of the data to the PDF of the estimated distribution:

Find the average annual minimum mean daily flow:

Find the probability that the minimum flow is 1.5 cubic meters per second or less:

Assuming that the annual minimum flows are independent, find the probability that the minimum flow will not exceed 2 cubic meters per second for 3 consecutive years:

Simulate annual minimum mean daily flows for the next 30 years:

MinStableDistribution can be used to model yield strength:

Fit MinStableDistribution into the data:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the average yield strength:

Find the probability that the yield strength is at least 38:

Simulate the yield strength for the next 50 samples:

MinStableDistribution can be used to model size. Consider particles of fly ash with diameters given in 20 microns:

Fit the distribution into the data:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the average particle diameter:

Find the probability that the diameter is at least 200 microns:

Simulate the diameters for 100 ash particles:

MinStableDistribution can be used to model the length of Cyrtoideae radiolarians:

Fit the distribution into the data:

Compare the histogram of the data with the PDF of the estimated distribution:

Find the average length of a cyrtoideae:

Find the probability that the length is at least 100 microns:

Simulate the lengths for 60 samples:

Properties & Relations (10)

Parameter influence on the CDF for each

:

MinStableDistribution is closed under translation and scaling by a positive factor:

CDF of MinStableDistribution solves the stability postulate equation:

Verify solution for

:

Find the limit of

:

Relationships to other distributions:

ExtremeValueDistribution is related to a generalized minimum extreme value distribution:

GumbelDistribution is a special case of a generalized minimum extreme value distribution:

Generalized minimum extreme value distribution is related to FrechetDistribution:

WeibullDistribution is a special case of a generalized minimum extreme value distribution:

Generalized minimum extreme value distribution is related to MaxStableDistribution:

ExpGammaDistribution is a special case of MinStableDistribution: