In probability theory, the Type-2 Gumbel probability density function is
f ( x | a , b ) = a b x − a − 1 e − b x − a {\displaystyle \ f(x|a,b)=a\ b\ x^{-a-1}\ e^{-b\ x^{-a}}\quad } for x > 0 . {\displaystyle \quad x>0~.}
For 0 < a ≤ 1 {\displaystyle \ 0<a\leq 1\ } the mean is infinite. For 0 < a ≤ 2 {\displaystyle \ 0<a\leq 2\ } the variance is infinite. The cumulative distribution function is
F ( x | a , b ) = e − b x − a . {\displaystyle \ F(x|a,b)=e^{-b\ x^{-a}}~.}
The moments E [ X k ] {\displaystyle \ \mathbb {E} {\bigl [}X^{k}{\bigr ]}\ } exist for k < a {\displaystyle \ k<a\ }
The distribution is named after Emil Julius Gumbel (1891 – 1966).
Generating random variates Given a random variate U {\displaystyle \ U\ } drawn from the uniform distribution in the interval ( 0 , 1 ) , {\displaystyle \ (0,1)\ ,} then the variate
X = ( − ln U b ) − 1 a {\displaystyle X=\left(-{\frac {\ln U}{b}}\right)^{-{\frac {1}{a}}}\ }
has a Type-2 Gumbel distribution with parameter a {\displaystyle \ a\ } and b . {\displaystyle \ b~.} This is obtained by applying the inverse transform sampling-method.
Related distributions The special case b = 1 {\displaystyle \ b=1\ } yields the Fréchet distribution. Substituting b = λ − k {\displaystyle \ b=\lambda ^{-k}\ } and a = − k {\displaystyle \ a=-k\ } yields the Weibull distribution. Note, however, that a positive k {\displaystyle \ k\ } (as in the Weibull distribution) would yield a negative a {\displaystyle \ a\ } and hence a negative probability density, which is not allowed. If X {\displaystyle X} is Type-2 Gumbel-distributed with parameters a {\displaystyle a} and b {\displaystyle b} , then X − 1 ∼ W e i b u l l ( a , b − 1 / a ) {\displaystyle X^{-1}\sim \mathrm {Weibull} (a,b^{-1/a})} .
Based on "Gumbel distribution". The GNU Scientific Library. type 002d2, used under GFDL.
See also Extreme value theory Gumbel distribution
