Parameters α
∈
(
0
,
∞
)
{displaystyle alpha in (0,infty )}
shape.
(Optionally, two more parameters)
s
∈
(
0
,
∞
)
{displaystyle sin (0,infty )}
scale (default:
s
=
1
{displaystyle s=1,}
)
m
∈
(
−
∞
,
∞
)
{displaystyle min (-infty ,infty )}
location of minimum (default:
m
=
0
{displaystyle m=0,}
) Support x
>
m
{displaystyle x>m} PDF α
s
(
x
−
m
s
)
−
1
−
α
e
−
(
x
−
m
s
)
−
α
{displaystyle {rac {alpha }{s}};left({rac {x-m}{s}}
ight)^{-1-alpha };e^{-({rac {x-m}{s}})^{-alpha }}} CDF e
−
(
x
−
m
s
)
−
α
{displaystyle e^{-({rac {x-m}{s}})^{-alpha }}} Mean {
m
+
s
Γ
(
1
−
1
α
)
for
α
>
1
∞
otherwise
{displaystyle {egin{cases} m+sGamma left(1-{rac {1}{alpha }}
ight)&{ ext{for }}alpha >1 infty &{ ext{otherwise}}end{cases}}} Median m
+
s
log
e
(
2
)
α
{displaystyle m+{rac {s}{sqrt[{alpha }]{log _{e}(2)}}}} |
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
Contents
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.
Characteristics
The single parameter Fréchet with parameter
(with
where
In particular:
The quantile
In particular the median is:
The mode of the distribution is
Especially for the 3-parameter Fréchet, the first quartile is
Also the quantiles for the mean and mode are: