Fréchet distribution
Encyclopedia
The Fréchet distribution is a special case of the generalized extreme value distribution. It has the cumulative distribution function
where α>0 is a shape parameter
. It can be generalised to include a location parameter
m (the minimum) and a scale parameter
s>0 with the cumulative distribution function
Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel
in 1958
has standardized moment
,
(with
) defined only for 
:
where
is the Gamma function
.
In particular:
The quantile
of order 
can be expressed through the inverse of the distribution,
.
In particular the median
is:
.
The mode
of the distribution is
.
Especially for the 3-parameter Fréchet, the first quartile is
and the third quartile
Also the quantiles for the mean and mode are:
where α>0 is a shape parameter
Shape parameter
In probability theory and statistics, a shape parameter is a kind of numerical parameter of a parametric family of probability distributions.- Definition :...
. It can be generalised to include a location parameter
Location parameter
In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution...
m (the minimum) and a scale parameter
Scale parameter
In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions...
s>0 with the cumulative distribution function
Named for Maurice Fréchet who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel
Emil Julius Gumbel
Emil Julius Gumbel was a Jewish mathematician and political writer.Born in Munich, he graduated from the University of Munich shortly before the outbreak of the First World War...
in 1958
Characteristics
The single parameter Fréchet with parameter has standardized moment,(with
) defined only for :where
is the Gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
.
In particular:
- For
the expectationExpectationIn the case of uncertainty, expectation is what is considered the most likely to happen. An expectation, which is a belief that is centered on the future, may or may not be realistic. A less advantageous result gives rise to the emotion of disappointment. If something happens that is not at all...
is
- For
the varianceVarianceIn probability theory and statistics, the variance is a measure of how far a set of numbers is spread out. It is one of several descriptors of a probability distribution, describing how far the numbers lie from the mean . In particular, the variance is one of the moments of a distribution...
is
.
The quantile
Quantile
Quantiles are points taken at regular intervals from the cumulative distribution function of a random variable. Dividing ordered data into q essentially equal-sized data subsets is the motivation for q-quantiles; the quantiles are the data values marking the boundaries between consecutive subsets...
of order can be expressed through the inverse of the distribution,.In particular the median
Median
In probability theory and statistics, a median is described as the numerical value separating the higher half of a sample, a population, or a probability distribution, from the lower half. The median of a finite list of numbers can be found by arranging all the observations from lowest value to...
is:
.The mode
Mode (statistics)
In statistics, the mode is the value that occurs most frequently in a data set or a probability distribution. In some fields, notably education, sample data are often called scores, and the sample mode is known as the modal score....
of the distribution is
.Especially for the 3-parameter Fréchet, the first quartile is
and the third quartileAlso the quantiles for the mean and mode are:
Applications
- In hydrologyHydrologyHydrology is the study of the movement, distribution, and quality of water on Earth and other planets, including the hydrologic cycle, water resources and environmental watershed sustainability...
, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. The blue picture illustrates an example of fitting the Fréchet distribution to ranked annually maximum one-day rainfalls in OmanOmanOman , officially called the Sultanate of Oman , is an Arab state in southwest Asia on the southeast coast of the Arabian Peninsula. It is bordered by the United Arab Emirates to the northwest, Saudi Arabia to the west, and Yemen to the southwest. The coast is formed by the Arabian Sea on the...
showing also the 90% confidence belt based on the binomial distribution. The cumulative frequencies of the rainfall data are represented by plotting positions as part of the cumulative frequency analysisCumulative frequency analysisCumulative frequency analysis is the applcation of estimation theory to exceedance probability . The complement, the non-exceedance probability concerns the frequency of occurrence of values of a phenomenon staying below a reference value. The phenomenon may be time or space dependent...
. However, in most hydrological applications, the distribution fitting is via the generalized extreme value distribution as this avoids imposing the assumption that the distribution does not have an upper bound (as required by the Frechet distribution when applied to annual maxima).
Related distributions
- If
(Uniform distribution (continuous)Uniform distribution (continuous)In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable. The support is defined by...
) then
- If
then 
- If
and 
then 
- The cumulative distribution functionCumulative distribution functionIn probability theory and statistics, the cumulative distribution function , or just distribution function, describes the probability that a real-valued random variable X with a given probability distribution will be found at a value less than or equal to x. Intuitively, it is the "area so far"...
of the Frechet distribution solves the maximum stability postulateStability postulateIn probability theory, to obtain a nondegenerate limiting distribution of the extreme value distribution, it is necessary to "reduce" the actual greatest value by applying a linear transformation with coefficients that depend on the sample size....
equation - If
(Weibull distribution) then 
Properties
- The Frechet distribution is a max stable distributionStability postulateIn probability theory, to obtain a nondegenerate limiting distribution of the extreme value distribution, it is necessary to "reduce" the actual greatest value by applying a linear transformation with coefficients that depend on the sample size....
- The negative of a random variable having a Frechet distribution is a min stable distributionStability postulateIn probability theory, to obtain a nondegenerate limiting distribution of the extreme value distribution, it is necessary to "reduce" the actual greatest value by applying a linear transformation with coefficients that depend on the sample size....
See also
- Type-2 Gumbel distribution
- Fisher–Tippett–Gnedenko theorem
- CumFreqCumFreqIn statistics and data analysis the application software CumFreq is a free and user-friendly tool for cumulative frequency analysis of a single variable and for probability distribution fitting....
(application software for probability distributions including Fréchet)
Publications
- Fréchet, M., (1927). "Sur la loi de probabilité de l'écart maximum." Ann. Soc. Polon. Math. 6, 93.
- Fisher, R.A., Tippett, L.H.C., (1928). "Limiting forms of the frequency distribution of the largest and smallest member of a sample." Proc. Cambridge Philosophical Society 24:180–190.
- Gumbel, E.J. (1958). "Statistics of Extremes." Columbia University Press, New York.
- Kotz, S.;Nadarajah, S. (2000) Extreme value distributions: theory and applications, World Scientific. ISBN 1860942245
