Probability density function
The Fréchet distribution is an absolutely continuous distribution over the positive real numbers that uses a really positive real scaling parameter . It is named after the French mathematician Maurice René Fréchet .
α
{\ displaystyle \ alpha}
Distribution and density function The Fréchet distribution has the distribution function for a real parameter > 0
α
{\ displaystyle {\ alpha}}
Φ
α
(
x
)
=
exp
(
-
x
-
α
)
=
exp
(
-
1
/
x
α
)
{\ displaystyle {\ Phi} _ {\ alpha} (x) = \ exp (-x ^ {- \ alpha}) = \ exp (-1 / x {^ {\ alpha}})}
The corresponding density function is
ϕ
α
(
x
)
=
α
x
-
(
α
+
1
)
e
-
x
-
α
{\ displaystyle {\ phi} _ {\ alpha} (x) = \ alpha \; x ^ {- (\ alpha +1)} \; e ^ {- x ^ {- \ alpha}}}
Moments and Median In the following we assume a -fréchet-distributed random variable and the gamma function .
X
{\ displaystyle X}
α
{\ displaystyle \ alpha}
Γ
(
x
)
{\ displaystyle \ Gamma \ left (x \ right)}
Median The median is
Med
(
X
)
=
(
1
log
e
(
2
)
)
1
/
α
{\ displaystyle \ operatorname {Med} (X) = \ left ({\ frac {1} {\ log _ {e} (2)}} \ right) ^ {1 / \ alpha}}
Existence of moments The kth moments of the Fréchet distribution exist if and only if .
α
>
k
{\ displaystyle \ alpha> k}
Expected value The expected value is
E.
(
X
)
=
Γ
(
1
-
1
α
)
{\ displaystyle \ operatorname {E} (X) = \ Gamma \ left (1 - {\ frac {1} {\ alpha}} \ right)}
Variance The variance is
Var
(
X
)
=
Γ
(
1
-
2
α
)
-
(
Γ
(
1
-
1
α
)
)
2
{\ displaystyle \ operatorname {Var} (X) = \ Gamma \ left (1 - {\ frac {2} {\ alpha}} \ right) - \ left (\ Gamma \ left (1 - {\ frac {1} {\ alpha}} \ right) \ right) ^ {2}}
Crookedness The crooked thing is
γ
m
(
X
)
=
Γ
(
1
-
3
α
)
-
3
Γ
(
1
-
2
α
)
Γ
(
1
-
1
α
)
+
2
Γ
3
(
1
-
1
α
)
(
Γ
(
1
-
2
α
)
-
Γ
2
(
1
-
1
α
)
)
3
2
{\ displaystyle \ gamma _ {m} (X) = {\ frac {\ Gamma \ left (1 - {\ frac {3} {\ alpha}} \ right) -3 \ Gamma \ left (1 - {\ frac {2} {\ alpha}} \ right) \ Gamma \ left (1 - {\ frac {1} {\ alpha}} \ right) +2 \ Gamma ^ {3} \ left (1 - {\ frac {1 } {\ alpha}} \ right)} {\ left (\ Gamma \ left (1 - {\ frac {2} {\ alpha}} \ right) - \ Gamma ^ {2} \ left (1 - {\ frac {1} {\ alpha}} \ right) \ right) ^ {\ frac {3} {2}}}}}
Kurtosis The kurtosis is
Kurt
(
X
)
=
-
6th
+
Γ
(
1
-
4th
α
)
-
4th
Γ
(
1
-
3
α
)
Γ
(
1
-
1
α
)
+
3
Γ
2
(
1
-
2
α
)
[
Γ
(
1
-
2
α
)
-
Γ
2
(
1
-
1
α
)
]
2
{\ displaystyle \ operatorname {Kurt} (X) = - 6 + {\ frac {\ Gamma \ left (1 - {\ frac {4} {\ alpha}} \ right) -4 \ Gamma \ left (1- { \ frac {3} {\ alpha}} \ right) \ Gamma \ left (1 - {\ frac {1} {\ alpha}} \ right) +3 \ Gamma ^ {2} \ left (1 - {\ frac {2} {\ alpha}} \ right)} {\ left [\ Gamma \ left (1 - {\ frac {2} {\ alpha}} \ right) - \ Gamma ^ {2} \ left (1- { \ frac {1} {\ alpha}} \ right) \ right] ^ {2}}}}
Relation to other distributions If Frèchet is distributed with parameters , then Gumbel is distributed with parameters and .
X
{\ displaystyle X}
α
{\ displaystyle \ alpha}
ln
X
{\ displaystyle \ ln X}
μ
=
0
{\ displaystyle \ mu = 0}
β
=
1
α
{\ displaystyle \ beta = {\ frac {1} {\ alpha}}}
According to Fisher-Tippet's theorem, a standardized, non-degenerate extreme value distribution can only converge to one of the three generalized extreme value distributions (GEV), one of which is the Fréchet distribution.
application It is therefore an important distribution for determining risks in financial statistics, such as value at risk and expected shortfall.
literature
J. Franke, W. Härdle, CM Hafner: Statistics of Financial Markets: An Introduction. 2nd Edition. Springer, Berlin / Heidelberg / New York 2008, ISBN 978-3-540-76269-0 .
J. Franke, CM Hafner, W. Härdle: Introduction to the statistics of the financial markets. 2nd Edition. Springer, Berlin / Heidelberg / New York 2004, ISBN 3-540-40558-5 .
Individual evidence
Discrete univariate distributions
Continuous univariate distributions
Multivariate distributions
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">
![{\ displaystyle \ operatorname {Kurt} (X) = - 6 + {\ frac {\ Gamma \ left (1 - {\ frac {4} {\ alpha}} \ right) -4 \ Gamma \ left (1- { \ frac {3} {\ alpha}} \ right) \ Gamma \ left (1 - {\ frac {1} {\ alpha}} \ right) +3 \ Gamma ^ {2} \ left (1 - {\ frac {2} {\ alpha}} \ right)} {\ left [\ Gamma \ left (1 - {\ frac {2} {\ alpha}} \ right) - \ Gamma ^ {2} \ left (1- { \ frac {1} {\ alpha}} \ right) \ right] ^ {2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b8c2d96f77fcae915045ec32d86e995a7375d2c)
