# MinStableDistribution

MinStableDistribution[μ,σ,ξ]

represents a generalized minimum extreme value distribution with location parameter μ, scale parameter σ, and shape parameter ξ.

# Details

• MinStableDistribution is also known as FisherTippett 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 μ and ξ to be any real numbers and σ to be any positive real number.
• MinStableDistribution allows μ and σ to be any quantities of any unit dimensions, and ξ to be a dimensionless quantity. »
• MinStableDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Background & Context

• MinStableDistribution[μ,σ,ξ] represents a continuous statistical distribution supported on the set of real numbers satisfying and parametrized by a positive real number σ (called a "scale parameter") and real numbers μ and ξ (a "location parameter" and a "shape parameter", respectively). Together, these parameters determine the overall behavior of its probability density function (PDF). In general, the PDF of a min-stable distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (height, spread, and horizontal location of its maximum) is determined by the values of μ, σ, and ξ. In addition, the tails of the PDF are "thin" in the sense that the PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) Along with the max-stable distribution (MaxStableDistribution), the min-stable distribution is a so-called "extreme value distribution" and may be referred to as the generalized minimum extreme value distribution, a type-1 extreme value distribution (not to be confused with ExtremeValueDistribution), a Gumbel minimum distribution (not to be confused with GumbelDistribution), and a Fisher-Tippett distribution.
• The generalized minimum extreme value distribution is the unique distribution modeling the asymptotic behavior of the minimum value in a sample from a distribution like the NormalDistribution, CauchyDistribution, or BetaDistribution, and was developed to combine the behaviors of other extreme value distributions such as GumbelDistribution, FrechetDistribution, and WeibullDistribution. Because its PDF is doubly exponential (i.e. is of the form Exp[-Exp[]]), the graph of the distribution has more exaggerated features (like higher peaks and thinner tails), a property unique among distributions. A cornerstone in the field known as extreme value theory, the min-stable distribution is widely utilized to describe situations that are "extremely unlikely" (i.e. those in which datasets consist of variates with extreme deviations from the median) and has been used to model a number of phenomena in various subfields of finance and economics.
• RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a min-stable distribution. Distributed[x,MinStableDistribution[μ,σ,ξ]], written more concisely as xMinStableDistribution[μ,σ,ξ], can be used to assert that a random variable x is distributed according to a min-stable distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
• The probability density and cumulative distribution functions may be given using PDF[MinStableDistribution[μ,σ,ξ],x] and CDF[MinStableDistribution[μ,σ,ξ],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
• DistributionFitTest can be used to test if a given dataset is consistent with a min-stable distribution, EstimatedDistribution to estimate a min-stable parametric distribution from given data, and FindDistributionParameters to fit data to a min-stable distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic extreme value distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic extreme value distribution.
• TransformedDistribution can be used to represent a transformed extreme value distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a min-stable distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving extreme value distributions.
• The min-stable distribution is related to a number of other distributions. MinStableDistribution generalizes a number of distributions including GumbelDistribution (GumbelDistribution[α,β] is the same as MinStableDistribution[α,β,0]), WeibullDistribution (WeibullDistribution[α,β] is precisely MinStableDistribution[β,β/α,-1/α]), and ExpGammaDistribution (ExpGammaDistribution[1,σ,μ] has the same PDF as MinStableDistribution[μ,σ,0]). It can be transformed to obtain the distribution functions for MaxStableDistribution, FrechetDistribution, and ExtremeValueDistribution. MinStableDistribution is also related to ExponentialDistribution and LogisticDistribution.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median:

## Scope(8)

Generate a sample of pseudorandom numbers from a 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:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the size quartiles:

## 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 to 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 kg/mm2:

Simulate the yield strength for 50 samples:

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

Fit the distribution to 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 to 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)

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

Scaling by a negative factor gives MaxStableDistribution:

MinStableDistribution is closed under taking Min:

Special case for shape parameter equal to 0:

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:

## Neat Examples(1)

PDFs for different ξ values with CDF contours:

Wolfram Research (2010), MinStableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MinStableDistribution.html (updated 2016).

#### Text

Wolfram Research (2010), MinStableDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MinStableDistribution.html (updated 2016).

#### CMS

Wolfram Language. 2010. "MinStableDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/MinStableDistribution.html.

#### APA

Wolfram Language. (2010). MinStableDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MinStableDistribution.html

#### BibTeX

@misc{reference.wolfram_2024_minstabledistribution, author="Wolfram Research", title="{MinStableDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/MinStableDistribution.html}", note=[Accessed: 23-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_minstabledistribution, organization={Wolfram Research}, title={MinStableDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/MinStableDistribution.html}, note=[Accessed: 23-June-2024 ]}