Gumbel distribution

The cumulative distribution function of the Gumbel distribution is

${\displaystyle F(x;\mu ,\beta )=e^{-e^{-(x-\mu )/\beta }}\,}$ 

The standard Gumbel distribution is the case where ${\displaystyle \mu =0}$  and ${\displaystyle \beta =1}$  with cumulative distribution function

${\displaystyle F(x)=e^{-e^{(-x)}}\,}$ 

and probability density function

${\displaystyle f(x)=e^{-(x+e^{-x})}.}$ 

In this case the mode is 0, the median is ${\displaystyle -\ln(\ln(2))\approx 0.3665}$ , the mean is ${\displaystyle \gamma \approx 0.5772}$  (the Euler–Mascheroni constant), and the standard deviation is ${\displaystyle \pi /{\sqrt {6}}\approx 1.2825.}$ 

The cumulants, for n > 1, are given by

${\displaystyle \kappa _{n}=(n-1)!\zeta (n).}$ 

The mode is μ, while the median is ${\displaystyle \mu -\beta \ln \left(\ln 2\right),}$  and the mean is given by

${\displaystyle \operatorname {E} (X)=\mu +\gamma \beta }$ ,

where ${\displaystyle \gamma }$  is the Euler–Mascheroni constant.

The standard deviation ${\displaystyle \sigma }$  is ${\displaystyle \beta \pi /{\sqrt {6}}}$  hence ${\displaystyle \beta =\sigma {\sqrt {6}}/\pi \approx 0.78\sigma .}$  ^[3]

At the mode, where ${\displaystyle x=\mu }$ , the value of ${\displaystyle F(x;\mu ,\beta )}$  becomes ${\displaystyle e^{-1}\approx 0.37}$ , irrespective of the value of ${\displaystyle \beta .}$ 

If ${\displaystyle G_{1},...,G_{k}}$  are iid Gumbel random variables with parameters ${\displaystyle (\mu ,\beta )}$  then ${\displaystyle \max\{G_{1},...,G_{k}\}}$  is also a Gumbel random variable with parameters ${\displaystyle (\mu +\beta \ln k,\beta )}$ .

If ${\displaystyle G_{1},G_{2},...}$  are iid random variables such that ${\displaystyle \max\{G_{1},...,G_{k}\}-\beta \ln k}$  has the same distribution as ${\displaystyle G_{1}}$  for all natural numbers ${\displaystyle k}$ , then ${\displaystyle G_{1}}$  is necessarily Gumbel distributed with scale parameter ${\displaystyle \beta }$  (actually it suffices to consider just two distinct values of k>1 which are coprime).

• If ${\displaystyle X}$  has a Gumbel distribution, then the conditional distribution of Y = −X given that Y is positive, or equivalently given that X is negative, has a Gompertz distribution. The cdf G of Y is related to F, the cdf of X, by the formula ${\displaystyle G(y)=P(Y\leq y)=P(X\geq -y\mid X\leq 0)=(F(0)-F(-y))/F(0)}$  for y > 0. Consequently, the densities are related by ${\displaystyle g(y)=f(-y)/F(0)}$ : the Gompertz density is proportional to a reflected Gumbel density, restricted to the positive half-line.^[4]
• If X is an exponentially distributed variable with mean 1, then −log(X) has a standard Gumbel distribution.
• If ${\displaystyle X\sim \mathrm {Gumbel} (\alpha _{X},\beta )}$  and ${\displaystyle Y\sim \mathrm {Gumbel} (\alpha _{Y},\beta )}$  are independent, then ${\displaystyle X-Y\sim \mathrm {Logistic} (\alpha _{X}-\alpha _{Y},\beta )\,}$  (see Logistic distribution).
• Despite this, if ${\displaystyle X,Y\sim \mathrm {Gumbel} (\alpha ,\beta )}$  are independent, then ${\displaystyle X+Y\nsim \mathrm {Logistic} (2\alpha ,\beta )}$ . This can easily be seen by noting that ${\displaystyle E(X+Y)=2\alpha +2\beta \gamma \neq 2\alpha =E\left(\mathrm {Logistic} (2\alpha ,\beta )\right)}$  (where ${\displaystyle \gamma }$  is the Euler-Mascheroni constant). Instead, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.^[5]

Theory related to the generalized multivariate log-gamma distribution provides a multivariate version of the Gumbel distribution.

 

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 ^[7] approaches the Gumbel distribution as the sample size increases.^[8]

Concretely, let ${\displaystyle \rho (x)=e^{-x}}$  be the probability distribution of ${\displaystyle x}$  and ${\displaystyle Q(x)=1-e^{-x}}$  its cumulative distribution. Then the maximum value out of ${\displaystyle N}$  realizations of ${\displaystyle x}$  is smaller than ${\displaystyle X}$  if and only if all realizations are smaller than ${\displaystyle X}$ . So the cumulative distribution of the maximum value ${\displaystyle {\tilde {x}}}$  satisfies

${\displaystyle P({\tilde {x}}-\log(N)\leq X)=P({\tilde {x}}\leq X+\log(N))=[Q(X+\log(N))]^{N}=\left(1-{\frac {e^{-X}}{N}}\right)^{N},}$ 

and, for large ${\displaystyle N}$ , the right-hand-side converges to ${\displaystyle e^{-e^{(-X)}}.}$ 

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,^[3] and also to describe droughts.^[9]

Gumbel has also shown that the estimator r⁄(n+1) 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^[10] as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.^[11]

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 "reparameterization tricks".^[12]

In detail, let ${\displaystyle (\pi _{1},\ldots ,\pi _{n})}$  be nonnegative, and not all zero, and let ${\displaystyle g_{1},\ldots ,g_{n}}$  be independent samples of Gumbel(0, 1), then by routine integration,$${\displaystyle Pr(j=\arg \max _{i}(g_{i}+\log \pi _{i}))={\frac {\pi _{j}}{\sum _{i}\pi _{i}}}}$$ That is, ${\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}$ 

Equivalently, given any ${\displaystyle x_{1},...,x_{n}\in \mathbb {R} }$ , we can sample from its Boltzmann distribution by

$${\displaystyle Pr(j=\arg \max _{i}(g_{i}+x_{i}))={\frac {e^{x_{j}}}{\sum _{i}e^{x_{i}}}}}$$ Related equations include:^[13]

• If ${\displaystyle x\sim \operatorname {Exp} (\lambda )}$ , then ${\displaystyle (-\ln x-\gamma )\sim {\text{Gumbel}}(-\gamma +\ln \lambda ,1)}$ .
• ${\displaystyle \arg \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Categorical}}\left({\frac {\pi _{j}}{\sum _{i}\pi _{i}}}\right)_{j}}$ .
• ${\displaystyle \max _{i}(g_{i}+\log \pi _{i})\sim {\text{Gumbel}}\left(\log \left(\sum _{i}\pi _{i}\right),1\right)}$ . That is, the Gumbel distribution is a max-stable distribution family.
• ${\displaystyle \mathbb {E} [\max _{i}(g_{i}+\beta x_{i})]=\log \left(\sum _{i}e^{\beta x_{i}}\right)+\gamma .}$ 

Since the quantile function (inverse cumulative distribution function), ${\displaystyle Q(p)}$ , of a Gumbel distribution is given by

${\displaystyle Q(p)=\mu -\beta \ln(-\ln(p)),}$ 

the variate ${\displaystyle Q(U)}$  has a Gumbel distribution with parameters ${\displaystyle \mu }$  and ${\displaystyle \beta }$  when the random variate ${\displaystyle U}$  is drawn from the uniform distribution on the interval ${\displaystyle (0,1)}$ .

 

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 ${\displaystyle F}$  :

${\displaystyle -\ln[-\ln(F)]={\frac {x-\mu }{\beta }}}$ 

In the paper the horizontal axis is constructed at a double log scale. The vertical axis is linear. By plotting ${\displaystyle F}$  on the horizontal axis of the paper and the ${\displaystyle x}$ -variable on the vertical axis, the distribution is represented by a straight line with a slope 1${\displaystyle /\beta }$ . When distribution fitting software like CumFreq became available, the task of plotting the distribution was made easier.

• Type-2 Gumbel distribution
• Extreme value theory
• Generalized extreme value distribution
• Fisher–Tippett–Gnedenko theorem
• Emil Julius Gumbel