The Fréchet distribution, also known as inverse Weibull 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 Maurice may refer to:
People
*Saint Maurice (died 287), Roman legionary and Christian martyr
*Maurice (emperor) or Flavius Mauricius Tiberius Augustus (539–602), Byzantine emperor
*Maurice (bishop of London) (died 1107), Lord Chancellor and Lo ...
who wrote a related paper in 1927, further work was done by Fisher and Tippett in 1928 and by Gumbel in 1958.
Characteristics
The single parameter Fréchet with parameter has standardized moment
:
(with ) defined only for :
:
where is the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
.
In particular:
* For the
expectation
Expectation or Expectations may refer to:
Science
* Expectation (epistemic)
* Expected value, in mathematical probability theory
* Expectation value (quantum mechanics)
* Expectation–maximization algorithm, in statistics
Music
* ''Expectation' ...
is
* For the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers ...
is
The
quantile
In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile tha ...
of order can be expressed through the inverse of the distribution,
:.
In particular the
median
In statistics and probability theory, the median is the value separating the higher half from the lower half of a data sample, a population, or a probability distribution. For a data set, it may be thought of as "the middle" value. The basic fe ...
is:
:
The
mode
Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to:
Arts and entertainment
* '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine
* ''Mode'' magazine, a fictional fashion magazine which is ...
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:
:
:
Applications
* In
hydrology
Hydrology () is the scientific study of the movement, distribution, and management of water on Earth and other planets, including the water cycle, water resources, and environmental watershed sustainability. A practitioner of hydrology is calle ...
, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. The blue picture, made with
CumFreq
In statistics and data analysis the application software CumFreq is a tool for cumulative frequency analysis of a single variable and for probability distribution fitting.
Originally the method was developed for the analysis of hydrological ...
, illustrates an example of fitting the Fréchet distribution to ranked annually maximum one-day rainfalls in
Oman
Oman ( ; ar, عُمَان ' ), officially the Sultanate of Oman ( ar, سلْطنةُ عُمان ), is an Arabian country located in southwestern Asia. It is situated on the southeastern coast of the Arabian Peninsula, and spans the mouth of t ...
showing also the 90%
confidence belt
In frequentist statistics, a confidence interval (CI) is a range of estimates for an unknown parameter. A confidence interval is computed at a designated ''confidence level''; the 95% confidence level is most common, but other levels, such as 9 ...
based on the
binomial distribution
In probability theory and statistics, the binomial distribution with parameters ''n'' and ''p'' is the discrete probability distribution of the number of successes in a sequence of ''n'' independent experiments, each asking a yes–no quest ...
. The cumulative frequencies of the rainfall data are represented by plotting positions as part of the
cumulative frequency analysis
Cumulative frequency analysis is the analysis of the frequency of occurrence of values of a phenomenon less than a reference value. The phenomenon may be time- or space-dependent. Cumulative frequency is also called ''frequency of non-exceedance ...
.
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 a lower bound (as required by the Frechet distribution).
* In
decline curve analysis
Decline curve analysis is a means of predicting future oil well or gas well production based on past production history. Production decline curve analysis is a traditional means of identifying well production problems and predicting well performan ...
, a declining pattern the time series data of oil or gas production rate over time for a well can be described by the Fréchet distribution.
* One test to assess whether a multivariate distribution is asymptotically dependent or independent consists of transforming the data into standard Fréchet margins using the transformation and then mapping from Cartesian to pseudo-polar coordinates . Values of correspond to the extreme data for which at least one component is large while approximately 1 or 0 corresponds to only one component being extreme.
Related distributions
*If (
Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions. The distribution describes an experiment where there is an arbitrary outcome that lies betwe ...
) then
*If then
*If and then
*The
cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x.
Ev ...