In probability theory and statistics, the Gumbel distribution (also known as the type-I generalized extreme value distribution) is used to model the distribution of the maximum (or the minimum) of a number of samples of various distributions.
This distribution might be used to represent the distribution of the maximum level of a river in a particular year if there was a list of maximum values for the past ten years. It is useful in predicting the chance that an extreme earthquake, flood or other natural disaster will occur. The potential applicability of the Gumbel distribution to represent the distribution of maxima relates to extreme value theory, which indicates that it is likely to be useful if the distribution of the underlying sample data is of the normal or exponential type.
The Gumbel distribution is a particular case of the generalized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as the log-Weibull distribution and the double exponential distribution (a term that is alternatively sometimes used to refer to the Laplace distribution). It is related to the Gompertz distribution: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.
In the latent variable formulation of the multinomial logit model — common in discrete choice theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributed random variables has a logistic distribution.
The Gumbel distribution is named after Emil Julius Gumbel (1891 - 1966), based on his original papers describing the distribution.[1]
The cumulative distribution function of the Gumbel distribution is
F(x;\mu,\beta)=
-e-(x-\mu)/\beta | |
e |
The standard Gumbel distribution is the case where
\mu=0
\beta=1
F(x)=
-e(-x) | |
e |
and probability density function
f(x)=
-(x+e-x) | |
e |
.
In this case the mode is 0, the median is
-ln(ln(2)) ≈ 0.3665
\gamma ≈ 0.5772
\pi/\sqrt{6} ≈ 1.2825.
The cumulants, for n > 1, are given by
\kappan=(n-1)!\zeta(n).
The mode is μ, while the median is
\mu-\betaln\left(ln2\right),
\operatorname{E}(X)=\mu+\gamma\beta
\gamma
The standard deviation
\sigma
\beta\pi/\sqrt{6}
\beta=\sigma\sqrt{6}/\pi ≈ 0.78\sigma.
At the mode, where
x=\mu
F(x;\mu,\beta)
e-1 ≈ 0.37
\beta.
If
G1,...,Gk
(\mu,\beta)
max\{G1,...,Gk\}
(\mu+\betalnk,\beta)
If
G1,G2,...
max\{G1,...,Gk\}-\betalnk
G1
k
G1
\beta
X
G(y)=P(Y\ley)=P(X\ge-y\midX\le0)=(F(0)-F(-y))/F(0)
g(y)=f(-y)/F(0)
X\simGumbel(\alphaX,\beta)
Y\simGumbel(\alphaY,\beta)
X-Y\simLogistic(\alphaX-\alphaY,\beta)
X,Y\simGumbel(\alpha,\beta)
X+Y\nsimLogistic(2\alpha,\beta)
E(X+Y)=2\alpha+2\beta\gamma ≠ 2\alpha=E\left(Logistic(2\alpha,\beta)\right)
Gumbel has shown that the maximum value (or last order statistic) in a sample of random variables following an exponential distribution minus the natural logarithm of the sample size [4] approaches the Gumbel distribution as the sample size increases.[5]
Concretely, let
\rho(x)=e-x
x
Q(x)=1-e-x
N
x
X
X
\tilde{x}
P(\tilde{x}-log(N)\leX)=P(\tilde{x}\leX+log(N))=[Q(X+log(N))]N=\left(1-
e-X | |
N |
\right)N,
N
-e(-X) | |
e |
.
In hydrology, therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,[6] and also to describe droughts.[7] Gumbel has also shown that the estimator for the probability of an event - where r is the rank number of the observed value in the data series and n is the total number of observations - is an unbiased estimator of the cumulative probability around the mode of the distribution. Therefore, this estimator is often used as a plotting position.
In number theory, the Gumbel distribution approximates the number of terms in a random partition of an integer[8] as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.[9]
It appears in the coupon collector's problem.
In machine learning, the Gumbel distribution is sometimes employed to generate samples from the categorical distribution. This technique is called "Gumbel-max trick" and is a special example of "reparametrization tricks".[10]
In detail, let
(\pi1,\ldots,\pin)
g1,\ldots,gn
\argmaxi(gi+log\pii)\simCategorical\left(
\pij | |
\sumi\pii |
\right)j
Equivalently, given any
x1,...,xn\in\R
Related equations include:[11]
x\sim\operatorname{Exp}(λ)
(-lnx-\gamma)\simGumbel(-\gamma+lnλ,1)
\argmaxi(gi+log\pii)\simCategorical\left(
\pij | |
\sumi\pii |
\right)j
maxi(gi+log\pii)\simGumbel\left(log\left(\sumi\pii\right),1\right)
E[maxi(gi+\betaxi)]=log\left(\sumi
\betaxi | |
e |
\right)+\gamma.
Since the quantile function (inverse cumulative distribution function),
Q(p)
Q(p)=\mu-\betaln(-ln(p)),
the variate
Q(U)
\mu
\beta
U
(0,1)
In pre-software times probability paper was used to picture the Gumbel distribution (see illustration). The paper is based on linearization of the cumulative distribution function
F
-ln[-ln(F)]=
x-\mu | |
\beta |
F
x
/\beta