Gumbel
Probability density function
Cumulative distribution function
Notation	${\displaystyle \text{Gumbel}(\mu ,\beta )}$
Parameters	${\displaystyle \mu ,}$location (real) ${\displaystyle \beta >0,}$scale (real)
Support	${\displaystyle x\in \mathbb{R}}$
PDF	${\displaystyle \frac{1}{\beta }{e}^{-(z+e^{-z})}}$ where${\displaystyle z=\frac{x-\mu }{\beta }}$
CDF	${\displaystyle e^{-e^{-(x-\mu )/\beta }}}$
Quantile	${\displaystyle \mu -\beta \ln (-\ln (p))}$
Mean	${\displaystyle \mu +\beta \gamma }$ where${\displaystyle \gamma }$ is theEuler–Mascheroni constant
Median	${\displaystyle \mu -\beta \ln (\ln 2)}$
Mode	${\displaystyle \mu }$
Variance	${\displaystyle \frac{{\pi }^2}{6}{\beta }^2}$
Skewness	${\displaystyle \frac{12\sqrt{6}\zeta (3)}{{\pi }^3}\approx 1.14}$
Excess kurtosis	${\displaystyle \frac{12}{5}}$
Entropy	${\displaystyle \ln (\beta )+\gamma +1}$
MGF	${\displaystyle \mathrm{\Gamma }(1-\beta t)e^{\mu t}}$
CF	${\displaystyle \mathrm{\Gamma }(1-i\beta t)e^{i\mu t}}$

Inprobability theory andstatistics, theGumbel distribution (also known as thetype-Igeneralized 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 toextreme 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.^[a]

The Gumbel distribution is a particular case of thegeneralized extreme value distribution (also known as the Fisher–Tippett distribution). It is also known as thelog-Weibull distribution and thedouble exponential distribution (a term that is alternatively sometimes used to refer to theLaplace distribution). It is related to theGompertz distribution: when its density is first reflected about the origin and then restricted to the positive half line, a Gompertz function is obtained.

In thelatent variable formulation of themultinomial logit model — common indiscrete choice theory — the errors of the latent variables follow a Gumbel distribution. This is useful because the difference of two Gumbel-distributedrandom variables has alogistic distribution.

The Gumbel distribution is named afterEmil Julius Gumbel (1891–1966), based on his original papers describing the distribution.^[1][2]

Thecumulative 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${\displaystyle \gamma \approx 0.5772}$ (theEuler–Mascheroni constant), and the standard deviation is${\displaystyle \pi /\sqrt{6}\approx \mathrm{1.2825.}}$

Thecumulants, forn > 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

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

where${\displaystyle \gamma }$ is theEuler–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).

Many problems indiscrete mathematics involve the study of an extremal parameter that follows a discrete version of the Gumbel distribution.^[4][5] Thisdiscrete version is the law of${\displaystyle Y=\lceil X\rceil }$, where${\displaystyle X}$ follows thecontinuous Gumbel distribution${\displaystyle \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu ,\beta )}$.Accordingly, this gives${\displaystyle P(Y\le h)=\exp (-\exp (-(h-\mu )/\beta ))}$ for any${\displaystyle h\in \mathbb{Z}}$.

Denoting${\displaystyle \mathrm{D}\mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu ,\beta )}$ as the discrete version, one has${\displaystyle \lceil X\rceil \sim \mathrm{D}\mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu ,\beta )}$ and${\displaystyle \lfloor X\rfloor \sim \mathrm{D}\mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu -1,\beta )}$.

There is no known closed form for the mean, variance (or higher-order moments) of the discrete Gumbel distribution, but it is easy to obtain high-precision numerical evaluations via rapidly converging infinite sums. For example, this yields${\displaystyle \mathbb{E}[\mathrm{D}\mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(0,1)]=\mathrm{1.077240905953631072609...}}$, but it remains an open problem to find a closed form for this constant (it is plausible there is none).

Aguech, Althagafi, and Banderier^[4] provide various bounds linking the discrete and continuous versions of the Gumbel distribution and explicitly detail (using methods fromMellin transform) the oscillating phenomena that appear when one has a sequence of random variables${\displaystyle \lfloor Y_n-c\ln n\rfloor }$ converging to a discrete Gumbel distribution.

• If${\displaystyle X}$ has a Gumbel distribution, then the conditional distribution of${\displaystyle Y=-X}$ given that${\displaystyle Y}$ is positive, or equivalently given that${\displaystyle X}$ is negative, has aGompertz distribution. The cdf${\displaystyle G}$ of${\displaystyle Y}$ is related to${\displaystyle F}$, the cdf of${\displaystyle X}$, by the formula${\displaystyle G(y)=P(Y\le y)=P(X\ge -y\mid X\le 0)=(F(0)-F(-y))/F(0)}$ for${\displaystyle y>0}$. Consequently, the densities are related by${\displaystyle g(y)=f(-y)/F(0)}$: theGompertz density is proportional to a reflected Gumbel density, restricted to the positive half-line.^[6]
• If${\displaystyle X\sim \mathrm{E}\mathrm{x}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}(1)}$ is anexponentially distributed variable with mean 1, then${\displaystyle \mu -\beta \log (X)\sim \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu ,\beta )}$.
• If${\displaystyle U\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}(0,1)}$ is auniformly distributed variable on the unit interval, then${\displaystyle \mu -\beta \log (-\log (U))\sim \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\mu ,\beta )}$.
• If${\displaystyle X\sim \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}({\alpha }_X,\beta )}$ and${\displaystyle Y\sim \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}({\alpha }_Y,\beta )}$ are independent, then${\displaystyle X-Y\sim \mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}({\alpha }_X-{\alpha }_Y,\beta )}$ (seeLogistic distribution).
• Despite this, if${\displaystyle X,Y\sim \mathrm{G}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{l}(\alpha ,\beta )}$ are independent, then${\displaystyle X+Y\nsim \mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}(2\alpha ,\beta )}$. This can easily be seen by noting that${\displaystyle \mathbb{E}(X+Y)=2\alpha +2\beta \gamma \ne 2\alpha =\mathbb{E}\left(\mathrm{L}\mathrm{o}\mathrm{g}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{i}\mathrm{c}(2\alpha ,\beta )\right)}$ (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.^[7]

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

Gumbel has shown that the maximum value (or lastorder statistic) in a sample ofrandom variables following anexponential distribution minus the natural logarithm of the sample size^[9] approaches the Gumbel distribution as the sample size increases.^[10]

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

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

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

Inhydrology, 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.^[11]

Gumbel has also shown that theestimatorr⁄(n+1) for the probability of an event — wherer is the rank number of the observed value in the data series andn is the total number of observations — is anunbiased estimator of thecumulative probability around themode of the distribution. Therefore, this estimator is often used as aplotting position.

• It is often of interest to predict probabilities out-of-sample data under the assumption that both the training data and the out-of-sample data follow a Gumbel distribution.
• Predictions of probabilities generated by substitutingmaximum likelihood estimates of the Gumbel parameters into thecumulative distribution function ignore parameter uncertainty. As a result, the probabilities are not wellcalibrated, do not reflect the frequencies of out-of-sample events, and, in particular, underestimate the probabilities of out-of-sample tail events.^[12]
• Predictions generated using the objectiveBayesian approach of calibrating prior prediction completely eliminate this underestimation. The Gumbel distribution is one of a number of statistical distributions withgroup structure, which arises because the Gumbel is alocation-scale model. As a result of the group structure, the Gumbel has associated left and rightHaar measures. The use of the rightHaar measure as theprior (known as the right Haar prior) in a Bayesian prediction gives probabilities that are perfectly calibrated, for any underlying true parameter values.^[13][12][14] Calibrating prior prediction for the Gumbel using the appropriate right Haar prior is implemented in theR software package fitdistcp.^[15]

Incombinatorics, the discrete Gumbel distribution appears as a limiting distribution for the hitting time in thecoupon collector's problem. This result was first established byLaplace in 1812 in hisThéorie analytique des probabilités, marking the first historical occurrence of what would later be called the Gumbel distribution.

Innumber theory, the Gumbel distribution approximates the number of terms in a randompartition of an integer^[16] as well as the trend-adjusted sizes of maximalprime gaps and maximal gaps betweenprime constellations.^[17]

Inprobability theory, it appears as the distribution of the maximum height reached by discrete walks (on the lattice${\displaystyle {\mathbb{N}}^2}$), where the process can be reset to its starting point at each step.^[4]

Inanalysis of algorithms, it appears, for example, in the study of the maximum carry propagation in base-${\displaystyle b}$ addition algorithms.^[18]

Since the quantile function (inversecumulative 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 theuniform 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${\displaystyle /\beta }$. Whendistribution fitting software like CumFreq became available, the task of plotting the distribution was made easier.

Inmachine learning, the Gumbel distribution is sometimes employed to generate samples from thecategorical distribution. This technique is called "Gumbel-max trick" and is a special example of "reparameterization tricks".^[19]

In detail, let${\displaystyle ({\pi }_1,\dots ,{\pi }_n)}$ be nonnegative, and not all zero, and let${\displaystyle g_1,\dots ,g_n}$ be independent samples of Gumbel(0, 1), then by routine integration,$${\displaystyle Pr(j=\arg \underset{i}{max}(g_i+\log {\pi }_i))=\frac{{\pi }_j}{\sum _i{\pi }_i}}$$That is,${\displaystyle \arg \underset{i}{max}(g_i+\log {\pi }_i)\sim \text{Categorical}{\left(\frac{{\pi }_j}{\sum _i{\pi }_i}\right)}_j}$

Equivalently, given any${\displaystyle x_1,...,x_n\in \mathbb{R}}$, we can sample from itsBoltzmann distribution by

$${\displaystyle Pr(j=\arg \underset{i}{max}(g_i+x_i))=\frac{e^{x_j}}{\sum _i{e}^{x_i}}}$$Related equations include:^[20]

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

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

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