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FrechetDistribution
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FrechetDistribution

✖

represents the Fréchet distribution with shape parameter α and scale parameter β.

✖

represents the Fréchet distribution with shape parameter α, scale parameter β, and location parameter μ.

Details

  • The Fréchet distribution gives the asymptotic distribution of the maximum value in a sample from a distribution such as the Cauchy distribution.
  • FrechetDistribution is also known as type II extreme value distribution.
  • The probability density for value in a Fréchet distribution is proportional to for and zero otherwise.
  • The probability density for value in a Fréchet distribution with location parameter is proportional to for and zero otherwise.
  • FrechetDistribution allows α and β to be any positive real numbers and μ to be any real number.
  • FrechetDistribution allows β and μ to be any quantities with the same unit dimensions and α to be a dimensionless quantity. »
  • FrechetDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

  • FrechetDistribution[α,β,μ] represents a continuous statistical distribution defined over the interval and parametrized by a real number μ (called a "location parameter") and two positive real numbers α and β (called a "shape parameter" and a "scale parameter", respectively). The probability density function (PDF) of a Fréchet distribution is unimodal, with the parameter μ determining the horizontal offset of the PDF and the parameters α and β together determining the PDF's overall height and steepness. In addition, the tails of the PDF are "thin", in the sense that the PDF decreases exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The Fréchet distribution is sometimes referred to as a type II extreme value distribution (not to be confused with "the" extreme value distribution, implemented in the Wolfram Language as ExtremeValueDistribution).
  • FrechetDistribution is one of four distributions (along with GumbelDistribution, ExtremeValueDistribution, and WeibullDistribution) classified under the general heading "extreme value distributions" and is therefore used as a tool for quantifying "extreme" or "rare" events. Named for French mathematician Maurice René Fréchet, the Fréchet distribution was introduced in the late 1920s as a possible limit distribution for the largest order statistic (i.e. as a potential asymptotic distribution for the maximum value of a sample distributed according to some other distribution). Since its inception, the Fréchet distribution has been linked to the modeling of a number of real-world phenomena, including human lifetimes, radioactive emissions, flood and seismic analyses, and maximum one-day rainfalls.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a Fréchet distribution. Distributed[x,FrechetDistribution[α,β,μ]], written more concisely as xFrechetDistribution[α,β,μ], can be used to assert that a random variable x is distributed according to a Fréchet 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[FrechetDistribution[α,β,μ],x] and CDF[FrechetDistribution[α,β,μ],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 Fréchet distribution, EstimatedDistribution to estimate a Fréchet parametric distribution from given data, and FindDistributionParameters to fit data to a Fréchet distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic Fréchet distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic Fréchet distribution.
  • TransformedDistribution can be used to represent a transformed Fréchet 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 Fréchet distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving Fréchet distributions.
  • The Fréchet distribution is related to several other distributions. As mentioned previously, FrechetDistribution shares qualitative relationships with ExtremeValueDistribution, GumbelDistribution, and WeibullDistribution. These relationships can be quantified by noting that the PDF of FrechetDistribution[α,β] is precisely the same as TransformedDistribution[β2/z,zWeibullDistribution[α,β]], and that WeibullDistribution can be written as transformed versions of both ExtremeValueDistribution and GumbelDistribution. The PDF of FrechetDistribution[α,β] is the same as MaxStableDistribution[β,β/α,1/α] and MinStableDistribution[-β,β/α,1/α] under suitable transformations and assumptions and can be obtained from UniformDistribution under a suitable chain of transformations. FrechetDistribution is also related to ExpGammaDistribution, ExponentialDistribution, and LogisticDistribution.

Examples

open allclose all

Basic Examples  (5)Summary of the most common use cases

Probability density function:

In[15]:=15
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https://wolfram.com/xid/0pnbht9zlx22hoka-8nv957
Out[15]=15
In[16]:=16
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https://wolfram.com/xid/0pnbht9zlx22hoka-09yruh
Out[16]=16
In[3]:=3
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https://wolfram.com/xid/0pnbht9zlx22hoka-dg16n
Out[3]=3

With location parameter:

In[12]:=12
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https://wolfram.com/xid/0pnbht9zlx22hoka-guvsip
Out[12]=12
In[5]:=5
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https://wolfram.com/xid/0pnbht9zlx22hoka-o45fre
Out[5]=5

Cumulative distribution function:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-ocpmrk
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-swhxzv
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-r8zsko
Out[3]=3

With location parameter:

In[4]:=4
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-haj0n0
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0pnbht9zlx22hoka-4ttbqv
Out[5]=5

Mean:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-8mh43
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0pnbht9zlx22hoka-7gjvmq
Out[2]=2

Variance:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-v0wyon
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0pnbht9zlx22hoka-tpwe0s
Out[2]=2

Median:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-6erttv
Out[1]=1
In[2]:=2
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https://wolfram.com/xid/0pnbht9zlx22hoka-i923rg
Out[2]=2

Scope  (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a Fréchet distribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-qhtk5j

Compare its histogram to the PDF:

In[3]:=3
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https://wolfram.com/xid/0pnbht9zlx22hoka-03mwaz
Out[3]=3

Distribution parameters estimation:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-epi747
Out[2]=2

Compare the density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-f8ui5o
Out[3]=3

Skewness depends only on the first parameter:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-rg8e7i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-s7jglt
Out[2]=2

Limiting values:

In[3]:=3
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https://wolfram.com/xid/0pnbht9zlx22hoka-828scd
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0pnbht9zlx22hoka-qp30pw
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-r6eqjl
Out[5]=5

Kurtosis depends only on the first parameter:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-zzxgo9
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-bz3hru
Out[2]=2

Limiting values:

In[3]:=3
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https://wolfram.com/xid/0pnbht9zlx22hoka-00e8xe
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0pnbht9zlx22hoka-hhqnw4
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-4tuwg0
Out[5]=5

Different moments with closed forms as functions of parameters:

In[1]:=1
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https://wolfram.com/xid/0pnbht9zlx22hoka-js043h

Moment:

In[2]:=2
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https://wolfram.com/xid/0pnbht9zlx22hoka-rx074o
Out[2]=2

Closed form for symbolic order:

In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-l5ytym
Out[3]=3

CentralMoment:

In[4]:=4
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-pknsqa
Out[4]=4

FactorialMoment:

In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-zg9ct4
Out[5]=5

Cumulant:

In[6]:=6
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-9gzmth
Out[6]=6

Hazard function:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-e2svfd
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-1xgftb
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-7t24wm
Out[3]=3

With location parameter:

In[4]:=4
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-t4upbb
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-wpdyuk
Out[5]=5

Quantile function:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-xnqfhj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-146fsh
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-6aiaj4
Out[3]=3

With location parameter:

In[4]:=4
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-dqi5us
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-yaor46
Out[5]=5

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-fsj0zx
Out[1]=1

Convert to milliliters:

In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-b4vv7f
Out[2]=2

Find the median:

In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-ehm63
Out[3]=3

Applications  (2)Sample problems that can be solved with this function

According to a study, the annual maximal tephra (solid material) volume in volcanic eruptions follows a FrechetDistribution with shape parameter 0.71 and scale parameter 6.3, given in cubic kilometers:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-viqmcx
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-8qrd2z
Out[2]=2

Find median annual maximal tephra volume:

In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-niwg3t
Out[3]=3

Find the average annual maximal tephra volume:

In[4]:=4
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https://wolfram.com/xid/0pnbht9zlx22hoka-1i762o
Out[4]=4

Find the mode of the distribution:

In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-86q2en
Out[5]=5

Find the probability that the annual maximal tephra volume is greater than 30 cubic kilometers:

In[6]:=6
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-ghf70w
Out[6]=6

Simulate the annual maximal tephra volume for the next 30 years:

In[7]:=7
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-ejs6yd
Out[7]=7

FrechetDistribution can be used to model annual maximum wind speeds:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-pcuqwr

Fit the distribution to the data:

In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-hanor9
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-fp21xa
Out[3]=3

Find the probability of annual maximum wind exceeding 90 km/h:

In[4]:=4
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-h4i4da
Out[4]=4

Find average annual maximum wind speed:

In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-ga35ko
Out[5]=5

Simulate maximum wind speed for 30 years:

In[6]:=6
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-lpe5mq
Out[6]=6

Properties & Relations  (8)Properties of the function, and connections to other functions

Fréchet distribution is closed under positive parameter scaling and translation:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-trasct
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-78ic4z
Out[2]=2

The family of FrechetDistribution is closed under maximum:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-2cnsmi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-byqsg1
Out[2]=2

CDF of FrechetDistribution solves the maximum stability postulate equation:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-lydrjv
Out[1]=1

Fix and and simplify:

In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-nu5644
Out[2]=2

Relationships to other distributions:

The default location is 0:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-lactu2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-8jg4be
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-5rwbo
Out[3]=3

Fréchet distribution is a transformation of WeibullDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-c0xn66
Out[1]=1

Fréchet distribution is related to MaxStableDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-gmn7q0
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-xgf8bo
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-iydq66
Out[3]=3

Fréchet distribution is related to MinStableDistribution:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-2itwt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-39oopn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-c92d5u
Out[3]=3

Neat Examples  (1)Surprising or curious use cases

PDFs for different α values with CDF contours:

In[1]:=1
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-io0104
In[5]:=5
✖
https://wolfram.com/xid/0pnbht9zlx22hoka-ceikus
Out[5]=5
Wolfram Research (2010), FrechetDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FrechetDistribution.html (updated 2016).
✖
Wolfram Research (2010), FrechetDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/FrechetDistribution.html (updated 2016).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_frechetdistribution, author="Wolfram Research", title="{FrechetDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/FrechetDistribution.html}", note=[Accessed: 09-May-2025 ]}

✖
@misc{reference.wolfram_2025_frechetdistribution, author="Wolfram Research", title="{FrechetDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/FrechetDistribution.html}", note=[Accessed: 09-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_frechetdistribution, organization={Wolfram Research}, title={FrechetDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/FrechetDistribution.html}, note=[Accessed: 09-May-2025 ]}

✖
@online{reference.wolfram_2025_frechetdistribution, organization={Wolfram Research}, title={FrechetDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/FrechetDistribution.html}, note=[Accessed: 09-May-2025 ]}