View source for Gumbel 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>
 |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>
}}
In [[probability theory]] and [[statistics]], the '''Gumbel distribution (Generalized Extreme Value distribution Type-I)''' 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. ''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 |format= 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}}</ref><ref>Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.</ref>

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

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

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\approx 0.5772</math> is the [[Euler&ndash;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> whatever the value of <math> \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</math>, and the standard deviation is <math>\pi/\sqrt{6} \approx 1.2825.</math>

The cumulants, for n>1, are given by 
:<math>\kappa_n = (n-1)! \zeta(n).</math>

==Quantile function and generating Gumbel variates==
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>.

==Relation to other distributions==
* If <math>X </math> has a Gumbel distribution, then the conditional distribution of ''Y=&minus;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 <math>G(y) = P(Y \le y) = P(X \ge -y | X \le 0) = (F(0)-F(-y))/F(0)</math> for ''y''>0. 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 }}</ref>
* If ''X'' is an exponentially distributed variable with mean 1, then &minus;log(''X'') has a standard Gumbel-Distribution.
* If <math>X \sim \mathrm{Gumbel}(\alpha_X, \beta) </math> and <math> Y \sim \mathrm{Gumbel}(\alpha_Y, \beta) </math> then <math> X-Y \sim \mathrm{Logistic}(\alpha_X-\alpha_Y,\beta) \,</math> (see [[Logistic distribution]]).
* If <math>X </math> and <math>Y \sim \mathrm{Gumbel}(\alpha, \beta) </math> then <math>X+Y \nsim \mathrm{Logistic}(2 \alpha,\beta) \,</math>. Note that <math> E(X+Y) = 2\alpha+2\beta\gamma \neq 2\alpha = E\left(\mathrm{Logistic}(2 \alpha,\beta) \right) </math>.
Theory related to the [[generalized multivariate log-gamma distribution]] provides a multivariate version of the Gumbel distribution.

==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)] = (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, as is demonstrated in the section below.

==Application==
[[File:FitGumbelDistr.tif|thumb|320px|[[Distribution fitting]] with [[confidence band]] of a cumulative Gumbel distribution to maximum one-day October rainfalls.<ref>[https://www.waterlog.info/cumfreq.htm CumFreq, software for probability distribution fitting]</ref> ]]

Gumbel has shown that the maximum value (or last [[order statistic]]) in a sample of a [[random variable]] following an [[exponential distribution]] approaches the Gumbel distribution closer with increasing sample size.<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>

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=175–224 |url=http://www.waterlog.info/pdf/freqtxt.pdf |isbn=90-70754-33-9}}</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 |pages=131 |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]].

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 [[machine learning]], the Gumbel distribution is sometimes employed to generate samples from the [[categorical distribution]].<ref>{{cite web |last1=Adams |first1=Ryan |title=The Gumbel-Max Trick for Discrete Distributions |url=http://lips.cs.princeton.edu/the-gumbel-max-trick-for-discrete-distributions/}}</ref>

==See also==
* [[Type-1 Gumbel distribution]]
* [[Type-2 Gumbel distribution]]
* [[Extreme value theory]]
* [[Generalized extreme value distribution]]
* [[Fisher–Tippett–Gnedenko theorem]]
* [[Emil Julius Gumbel]]

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

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

{{ProbDistributions|continuous-infinite}}

{{DEFAULTSORT:Gumbel Distribution}}
[[Category:Continuous distributions]]
[[Category:Extreme value data]]
[[Category:Location-scale family probability distributions]]