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
Pr
(
X
≤
x
)
=
e
−
x
−
α
if
x
>
0
.
{\displaystyle \ \Pr(\ X\leq x\ )=e^{-x^{-\alpha }}~{\text{ if }}~x>0~.}
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
Pr
(
X
≤
x
)
=
exp
[
−
(
x
−
m
s
)
−
α
]
if
x
>
m
.
{\displaystyle \ \Pr(\ X\leq x\ )=\exp \left[\ -\left({\tfrac {\ x-m\ }{s}}\right)^{-\alpha }\ \right]~~{\text{ if }}~x>m~.}
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.
Contents
Characteristics
The single parameter Fréchet, with parameter
α
{\displaystyle \ \alpha }
, has standardized moment
μ
k
=
∫
0
∞
x
k
f
(
x
)
d
x
=
∫
0
∞
t
−
k
α
e
−
t
d
t
,
{\displaystyle \mu _{k}=\int _{0}^{\infty }x^{k}f(x)\ \operatorname {d} x=\int _{0}^{\infty }t^{-{\frac {k}{\alpha }}}e^{-t}\ \operatorname {d} t\ ,}
Properties
The Frechet distribution is a max stable distribution
The negative of a random variable having a Frechet distribution is a min stable distribution
Related distributions
The cumulative distribution function of the Frechet distribution solves the maximum stability postulate equation.
Scaling relations include:
If
X
∼
U
(
0
,
1
)
{\displaystyle \ X\sim U(\ 0,1\ )\ }
(continuous uniform distribution) then
m
+
s
⋅
(
−
log
e
(
X
)
)
−
1
α
∼
Frechet
(
α
,
Applications
In hydrology, the Fréchet distribution is applied to extreme events such as annually maximum one-day rainfalls and river discharges. This picture illustrates an example of fitting the Fréchet distribution to ranked annually maximum one-day rainfalls in Oman 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 analysis. 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, 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
Z
i
=
−
1
/
log
F
i
(
X
i
)
{\displaystyle Z_{i}=-1/\log F_{i}(X_{i})}
and then mapping from Cartesian to pseudo-polar coordinates
(