Editing Gumbel distribution

{{Short description|Particular case of the generalized extreme value distribution}}
{{Probability distribution 
 |name       =Gumbel
 |type       =density
 |pdf_image  =[[File:Gumbel-Density.svg|325px|Probability distribution function]]
 |cdf_image  =[[File:Gumbel-Cumulative.svg|325px|Cumulative distribution function]]
 |parameters =<math>\mu,</math> [[location parameter|location]] ([[real numbers|real]])<br /><math>\beta>0,</math> [[scale parameter|scale]] (real)
 |support    =<math>x\in\mathbb{R}</math>
 |pdf        =<math>\frac{1}{\beta}e^{-(z+e^{-z})}</math><br /> where <math>z=\frac{x-\mu}{\beta}</math>
 |cdf        =<math>e^{-e^{-(x-\mu)/\beta}}</math>
 |quantile   =<math>\mu-\beta\ln(-\ln(p))</math>
 |mean       =<math>\mu + \beta\gamma</math> <br> where <math>\gamma</math> is the [[Euler&ndash;Mascheroni constant]]
 |median     =<math>\mu - \beta\ln(\ln 2)</math>
 |mode       =<math>\mu</math>
 |variance   =<math>\frac{\pi^2}{6}\beta^2</math>
 |skewness   =<math>\frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14</math>
 |kurtosis   =<math>\frac{12}{5}</math>
 |entropy    =<math>\ln(\beta)+\gamma+1</math>
 |mgf        =<math>\Gamma(1-\beta t) e^{\mu t}</math>
 |char       =<math>\Gamma(1-i\beta t) e^{i\mu t}</math>
|notation=<math>\text{Gumbel}(\mu, \beta)</math>}}
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.{{efn|This article uses the Gumbel distribution to model the distribution of the maximum value''. ''To model the minimum  value, use the negative of the original values.}}

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 variable]]s has a [[logistic distribution]].

The Gumbel distribution is named after [[Emil Julius Gumbel]] (1891&ndash;1966), based on his original papers describing the distribution.<ref>{{Citation |url= http://archive.numdam.org/article/AIHP_1935__5_2_115_0.pdf |title= Les valeurs extrêmes des distributions statistiques |last= Gumbel |first= E.J. |journal=  Annales de l'Institut Henri Poincaré |volume= 5 |year=  1935 |pages= 115–158 |issue= 2 |author-link= Emil Julius Gumbel}}</ref><ref>Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.</ref>

==Definitions==
The [[cumulative distribution function]] of the Gumbel distribution is

:<math>F(x;\mu,\beta) = e^{-e^{-(x-\mu)/\beta}}\,</math>

===Standard Gumbel distribution===
The standard Gumbel distribution is the case where <math>\mu = 0</math> and <math>\beta = 1</math> with cumulative distribution function
:<math>F(x) = e^{-e^{-x}}\,</math>

and probability density function  
:<math>f(x) = e^{-(x+e^{-x})}.</math>

In this case the mode is 0, the median is <math>-\ln(\ln(2)) \approx 0.3665</math>, the mean is <math>\gamma\approx 0.5772</math> (the [[Euler&ndash;Mascheroni constant]]), and the standard deviation is <math>\pi/\sqrt{6} \approx 1.2825.</math>

The [[Cumulant|cumulants]], for ''n''&nbsp;>&nbsp;1, are given by 
:<math>\kappa_n = (n-1)! \zeta(n).</math>

==Properties==
The mode is μ, while the median is <math>\mu-\beta \ln\left(\ln 2\right),</math> and the mean is given by
:<math>\operatorname{E}(X)=\mu+\gamma\beta</math>,
where <math> \gamma </math> is the [[Euler–Mascheroni constant]].

The standard deviation <math> \sigma </math> is <math>\beta \pi/\sqrt{6}</math> hence <math>\beta = \sigma \sqrt{6} / \pi \approx 0.78 \sigma. </math> <ref name = "Oosterbaan" />

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

If <math>G_1,...,G_k</math> are iid Gumbel random variables with parameters <math>(\mu,\beta)</math> then <math>\max\{G_1,...,G_k\}</math> is also a Gumbel random variable with parameters <math>(\mu+\beta\ln k, \beta)</math>. 

If <math>G_1, G_2,...</math> are iid random variables such that <math>\max\{G_1,...,G_k\}-\beta\ln k </math> has the same distribution as <math>G_1</math> for all natural numbers <math> k </math>, then <math>G_1</math> is necessarily Gumbel distributed with scale parameter <math>\beta</math> (actually it suffices to consider just two distinct values of k>1 which are coprime).

==Related distributions==
===The discrete Gumbel distribution===
Many problems in [[discrete mathematics]] involve the study of an extremal parameter that follows a discrete version of the Gumbel distribution.<ref name=AguechAlthagafiBanderier>
{{Citation |arxiv=2311.13124|title=Height of walks with resets, the Moran model, and the discrete Gumbel distribution  |year=2023|first1=R.|last1=Aguech|first2=A.|last2=Althagafi|first3=C.|last3=Banderier|journal=Séminaire Lotharingien de Combinatoire|volume=87B|issue=12|pages=1–37}}</ref><ref>''Analytic Combinatorics'', Flajolet and Sedgewick.</ref> This ''discrete'' version is the law of <math>Y = \lceil X \rceil</math>, where <math>X</math> follows the ''continuous'' Gumbel distribution <math>\mathrm{Gumbel}(\mu, \beta)</math>.
Accordingly, this gives <math>P(Y \leq h) = \exp(-\exp(-(h-\mu)/\beta))</math> for any <math>h \in \mathbb Z</math>.

Denoting <math>\mathrm{DGumbel}(\mu, \beta)</math> as the discrete version, one has <math>\lceil X \rceil \sim \mathrm{DGumbel}(\mu, \beta)</math> and <math>\lfloor X \rfloor \sim \mathrm{DGumbel}(\mu - 1, \beta)</math>.

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 <math>{\mathbb E}[\mathrm{DGumbel}(0,1)]=1.077240905953631072609...</math>, but it remains an open problem to find a closed form for this constant (it is plausible there is none).

Aguech, Althagafi, and Banderier<ref name=AguechAlthagafiBanderier/> provide various bounds linking the discrete and continuous versions of the Gumbel distribution and explicitly detail (using methods from [[Mellin transform]]) the oscillating phenomena that appear when one has a sequence of random variables <math>\lfloor Y_n - c \ln n \rfloor</math> converging to a discrete Gumbel distribution.

===Continuous distributions===
* If <math>X</math> has a Gumbel distribution, then the conditional distribution of <math>Y=-X</math> given that <math>Y</math> is positive, or equivalently given that <math>X</math> is negative, has a [[Gompertz distribution]]. The cdf <math>G</math> of <math>Y</math> is related to <math>F</math>, the cdf of <math>X</math>, by the formula <math>G(y) = P(Y \le y) = P(X \ge -y \mid X \le 0) = (F(0)-F(-y))/F(0)</math> for <math>y>0</math>. Consequently, the densities are related by <math>g(y) = f(-y)/F(0)</math>: the [[Gompertz function|Gompertz density]] is proportional to a reflected Gumbel density, restricted to the positive half-line.<ref>{{Cite journal |doi=10.1016/j.insmatheco.2006.07.003 |title=Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz' law of mortality |year=2007 |last1=Willemse |first1=W.J. |last2=Kaas |first2=R. |journal=Insurance: Mathematics and Economics |volume=40 |issue=3 |pages=468 |url=https://www.dnb.nl/binaries/Working%20Paper%20135-2007_tcm46-146792.pdf |access-date=2019-09-24 |archive-date=2017-08-09 |archive-url=https://web.archive.org/web/20170809050854/https://www.dnb.nl/binaries/Working%20Paper%20135-2007_tcm46-146792.pdf |url-status=dead }}</ref>
* If <math>X\sim\mathrm{Exponential}(1)</math> is an [[Exponential distribution|exponentially distributed]] variable with mean 1, then <math>\mu -\beta\log(X)\sim\mathrm{Gumbel}(\mu,\beta)</math>.
* If <math>U\sim\mathrm{Uniform}(0,1)</math> is a [[Continuous uniform distribution|uniformly distributed]] variable on the unit interval, then <math> \mu -\beta\log(-\log(U))\sim\mathrm{Gumbel}(\mu,\beta)</math>.
* If <math>X \sim \mathrm{Gumbel}(\alpha_X, \beta) </math> and <math> Y \sim \mathrm{Gumbel}(\alpha_Y, \beta) </math> are independent, then <math> X-Y \sim \mathrm{Logistic}(\alpha_X-\alpha_Y,\beta) \,</math> (see [[Logistic distribution]]).
* Despite this, if <math>X, Y \sim \mathrm{Gumbel}(\alpha, \beta) </math> are independent, then <math>X+Y \nsim \mathrm{Logistic}(2 \alpha,\beta)</math>. This can easily be seen by noting that <math>\mathbb{E}(X+Y) = 2\alpha+2\beta\gamma \neq 2\alpha = \mathbb{E}\left(\mathrm{Logistic}(2 \alpha,\beta) \right) </math> (where <math>\gamma</math> is the Euler-Mascheroni constant). Instead, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.<ref name="Marques">{{Cite journal | last1=Marques|first1 = F.|  last2=Coelho| first2=C.| last3=de Carvalho|first3=M.| title = On the distribution of linear combinations of independent Gumbel random variables | journal=Statistics and Computing|year=2015|volume=25 | issue=3 | pages=683‒701| doi=10.1007/s11222-014-9453-5 | s2cid=255067312 | url=https://www.maths.ed.ac.uk/~mdecarv/papers/marques2015.pdf}}</ref>

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

==Occurrence and applications==
===Applications of the continous Gumbel distribution===
[[File:FitGumbelDistr.tif|thumb|320px|[[Distribution fitting]] with [[confidence band]] of a cumulative Gumbel distribution to maximum one-day October rainfalls.<ref>{{Cite web|url=https://www.waterlog.info/cumfreq.htm|title=CumFreq, distribution fitting of probability, free calculator|website=www.waterlog.info}}</ref> ]]

Gumbel has shown that the maximum value (or last [[order statistic]]) in a sample of [[random variable]]s following an [[exponential distribution]] minus the natural logarithm of the sample size <ref>{{Cite web|url=https://math.stackexchange.com/questions/3527556/gumbel-distribution-and-exponential-distribution?noredirect=1#comment7669633_3527556|title=Gumbel distribution and exponential distribution|website=Mathematics Stack Exchange}}</ref>  approaches the Gumbel distribution as the sample size increases.<ref>{{cite book |last=Gumbel |first= E.J. |year=1954 |asin=B0007DSHG4 |title=Statistical theory of extreme values and some practical applications |series=Applied Mathematics Series |volume= 33 |edition=1st |url= https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/PB175818.xhtml |publisher= U.S. Department of Commerce, National Bureau of Standards}}</ref>

Concretely, let <math> \rho(x)=e^{-x} </math> be the probability distribution of <math> x </math> and <math> Q(x)=1- e^{-x} </math> its cumulative distribution. Then the maximum value out of <math> N </math> realizations of <math> x </math> is smaller than <math> X </math> if and only if all realizations are smaller than <math> X </math>. So the cumulative distribution of the maximum value <math> \tilde{x} </math> satisfies
:<math>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, </math>
and, for large <math> N </math>, the right-hand-side converges to <math> e^{-e^{(-X)}}. </math>

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,<ref name = "Oosterbaan">{{cite book |editor-last=Ritzema |editor-first=H.P. |first1=R.J. |last1=Oosterbaan |chapter=Chapter 6 Frequency and Regression Analysis |year=1994 |title=Drainage Principles and Applications, Publication 16 |publisher=International Institute for Land Reclamation and Improvement (ILRI) |location=Wageningen, The Netherlands |pages=[https://archive.org/details/drainageprincipl0000unse/page/175 175–224] |chapter-url=http://www.waterlog.info/pdf/freqtxt.pdf |isbn=90-70754-33-9 |url=https://archive.org/details/drainageprincipl0000unse/page/175 }}</ref> and also to describe droughts.<ref>{{cite journal |doi=10.1016/j.jhydrol.2010.04.035 |title=An extreme value analysis of UK drought and projections of change in the future |year=2010 |last1=Burke |first1=Eleanor J. |last2=Perry |first2=Richard H.J. |last3=Brown |first3=Simon J. |journal=Journal of Hydrology |volume=388 |issue=1–2 |pages=131–143 |bibcode=2010JHyd..388..131B}}</ref>
 
Gumbel has also shown that the [[estimator]] {{frac|''r''|(''n''+1)}} for the probability of an event &mdash; where ''r'' is the rank number of the observed value in the data series and ''n'' is the total number of observations &mdash; is an [[unbiased estimator]] of the [[cumulative probability]] around the [[Mode (statistics)|mode]] of the distribution. Therefore, this estimator is often used as a [[plotting position]].

===Occurrences of the discrete Gumbel distribution===
In [[combinatorics]], the discrete Gumbel distribution appears as a limiting distribution for the hitting time in the [[coupon collector's problem]]. This result was first established by [[Pierre-Simon de Laplace|Laplace]] in 1812 in his ''Théorie analytique des probabilités'', marking the first historical occurrence of what would later be called the Gumbel distribution.

In [[number theory]], the Gumbel distribution approximates the number of terms in a random [[partition of an integer]]<ref>{{cite journal |doi=10.1215/S0012-7094-41-00826-8 |title=The distribution of the number of summands in the partitions of a positive integer |year=1941 |last1=Erdös |first1=Paul |last2=Lehner |first2=Joseph |journal=Duke Mathematical Journal |volume=8 |issue=2 |pages=335}}</ref> as well as the trend-adjusted sizes of maximal [[prime gaps]] and maximal gaps between [[prime constellations]].<ref>{{cite journal |arxiv=1301.2242 |last=Kourbatov |first= A. |title=Maximal gaps between prime ''k''-tuples: a statistical approach |journal=Journal of Integer Sequences |volume=16 |year=2013|bibcode=2013arXiv1301.2242K }} Article 13.5.2.</ref>

In [[probability theory]], it appears as the distribution of the maximum height reached by discrete walks (on the lattice <math>{\mathbb N}^2</math>), where the process can be reset to its starting point at each step.<ref name=AguechAlthagafiBanderier/>

In [[analysis of algorithms]], it appears, for example, in the study of the maximum carry propagation in base-<math>b</math> addition algorithms.<ref>{{citation |title=The average time for carry propagation|year=1978|first1=Donald E.|last1=Knuth|journal=Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagationes Mathematicae|volume=81|pages=238–242}}</ref>

==Random variate generation==
{{further|Non-uniform random variate generation}}

Since the quantile function (inverse [[cumulative distribution function]]), <math>Q(p)</math>, of a Gumbel distribution is given by

:<math>Q(p)=\mu-\beta\ln(-\ln(p)),</math>

the variate <math>Q(U)</math> has a Gumbel distribution with parameters <math>\mu</math> and <math>\beta</math> when the random variate <math>U</math> is drawn from the [[uniform distribution (continuous)|uniform distribution]] on the interval <math>(0,1)</math>.

===Probability paper===
[[File:Gumbel paper.JPG|thumb|320px|A piece of graph paper that incorporates the Gumbel distribution.]]

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 <math>F</math> :
: <math> -\ln(-\ln(F)) = \frac{x-\mu}\beta </math>
In the paper the horizontal axis is constructed at a double log scale. The vertical axis is linear. By plotting <math>F</math> on the horizontal axis of the paper and the <math>x</math>-variable on the vertical axis, the distribution is represented by a straight line with a slope 1<math>/\beta</math>. When [[distribution fitting]] software like [[CumFreq]] became available, the task of plotting the distribution was made easier.

=== Gumbel reparameterization tricks ===
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 trick|reparameterization tricks]]".<ref>{{Cite conference |first1=Eric |last1=Jang |first2=Shixiang |last2=Gu |first3=Ben |last3=Poole |date=April 2017 |title=Categorical Reparameterization with Gumble-Softmax |url=https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_2564872 |conference=International Conference on Learning Representations (ICLR) 2017}}</ref>

In detail, let <math>(\pi_1, \ldots, \pi_n)</math> be nonnegative, and not all zero, and let <math>g_1,\ldots , g_n</math> be independent samples of Gumbel(0, 1), then by routine integration,<math display="block">Pr(j = \arg\max_i (g_i + \log\pi_i)) = \frac{\pi_j}{\sum_i \pi_i}</math>That is, <math>\arg\max_i (g_i + \log\pi_i) \sim \text{Categorical}\left(\frac{\pi_j}{\sum_i \pi_i}\right)_j</math>

Equivalently, given any <math>x_1, ..., x_n\in \R</math>, we can sample from its [[Boltzmann distribution]] by

<math display="block">Pr(j = \arg\max_i (g_i + x_i)) = \frac{e^{x_j}}{\sum_i e^{x_i}}</math>Related equations include:<ref>{{Cite journal |last1=Balog |first1=Matej |last2=Tripuraneni |first2=Nilesh |last3=Ghahramani |first3=Zoubin |last4=Weller |first4=Adrian |date=2017-07-17 |title=Lost Relatives of the Gumbel Trick |url=https://proceedings.mlr.press/v70/balog17a.html |journal=International Conference on Machine Learning |language=en |publisher=PMLR |pages=371–379|arxiv=1706.04161 }}</ref>
* If <math>x\sim \operatorname{Exp}(\lambda)</math>, then <math>(-\ln x - \gamma)\sim \text{Gumbel}(-\gamma + \ln\lambda, 1)</math>.
* <math>\arg\max_i (g_i + \log\pi_i) \sim \text{Categorical}\left(\frac{\pi_j}{\sum_i \pi_i}\right)_j</math>.
* <math>\max_i (g_i + \log\pi_i) \sim \text{Gumbel}\left(\log\left(\sum_i \pi_i \right), 1\right)</math>. That is, the Gumbel distribution is a max-stable distribution family.
* <math>\mathbb E[\max_i (g_i + \beta x_i)] = \log \left(\sum_i e^{\beta x_i}\right) + \gamma.</math>


==See also==
* [[Type-2 Gumbel distribution]]
* [[Extreme value theory]]
* [[Generalized extreme value distribution]]
* [[Fisher–Tippett–Gnedenko theorem]]
* [[Emil Julius Gumbel]]

==Notes==
{{Notelist}}

==References==
{{Reflist|40em}}

== External links ==
{{Commons and category}}

{{ProbDistributions|continuous-infinite}}

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[[Category:Continuous distributions]]
[[Category:Extreme value data]]
[[Category:Location-scale family probability distributions]]