Gumbel distribution

The cumulative distribution function of the Gumbel distribution is

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

The standard Gumbel distribution is the case where $\backslash mu\; =\; 0$ and $\backslash beta\; =\; 1$ with cumulative distribution function

:$F(x)\; =\; e^\{-e^\{-x\}\}\backslash ,$

and probability density function

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

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

The cumulants, for n > 1, are given by

:$\backslash kappa\_n\; =\; (n-1)!\; \backslash zeta(n).$

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

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

where $\backslash gamma$ is the Euler–Mascheroni constant.

The standard deviation $\backslash sigma$ is $\backslash beta\; \backslash pi/\backslash sqrt\{6\}$ hence $\backslash beta\; =\; \backslash sigma\; \backslash sqrt\{6\}\; /\; \backslash pi\; \backslash approx\; 0.78\; \backslash sigma.$

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

If $G\_1,...,G\_k$ are iid Gumbel random variables with parameters $(\backslash mu,\backslash beta)$ then $\backslash max\backslash \{G\_1,...,G\_k\backslash \}$ is also a Gumbel random variable with parameters $(\backslash mu+\backslash beta\backslash ln\; k,\; \backslash beta)$.

If $G\_1,\; G\_2,...$ are iid random variables such that $\backslash max\backslash \{G\_1,...,G\_k\backslash \}-\backslash beta\backslash ln\; k$ has the same distribution as $G\_1$ for all natural numbers $k$, then $G\_1$ is necessarily Gumbel distributed with scale parameter $\backslash beta$ (actually it suffices to consider just two distinct values of k>1 which are coprime).

Many problems in discrete mathematics involve the study of an extremal parameter that follows a discrete version of the Gumbel distribution.

{{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}}Analytic Combinatorics, Flajolet and Sedgewick. This discrete version is the law of $Y\; =\; \backslash lceil\; X\; \backslash rceil$, where $X$ follows the continuous Gumbel distribution $\backslash mathrm\{Gumbel\}(\backslash mu,\; \backslash beta)$.

Accordingly, this gives $P(Y\; \backslash leq\; h)\; =\; \backslash exp(-\backslash exp(-(h-\backslash mu)/\backslash beta))$ for any $h\; \backslash in\; \backslash mathbb\; Z$.

Denoting $\backslash mathrm\{DGumbel\}(\backslash mu,\; \backslash beta)$ as the discrete version, one has $\backslash lceil\; X\; \backslash rceil\; \backslash sim\; \backslash mathrm\{DGumbel\}(\backslash mu,\; \backslash beta)$ and $\backslash lfloor\; X\; \backslash rfloor\; \backslash sim\; \backslash mathrm\{DGumbel\}(\backslash mu\; -\; 1,\; \backslash beta)$.

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

Aguech, Althagafi, and Banderier 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 $\backslash lfloor\; Y\_n\; -\; c\; \backslash ln\; n\; \backslash rfloor$ converging to a discrete Gumbel distribution.

• If $X$ 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 $G(y)\; =\; P(Y\; \backslash le\; y)\; =\; P(X\; \backslash ge\; -y\; \backslash mid\; X\; \backslash le\; 0)\; =\; (F(0)-F(-y))/F(0)$ for $y>0$. Consequently, the densities are related by $g(y)\; =\; f(-y)/F(0)$: the Gompertz density is proportional to a reflected Gumbel density, restricted to the positive half-line.{{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 }}
• If $X\backslash sim\backslash mathrm\{Exponential\}(1)$ is an exponentially distributed variable with mean 1, then $\backslash mu\; -\backslash beta\backslash log(X)\backslash sim\backslash mathrm\{Gumbel\}(\backslash mu,\backslash beta)$.
• If $U\backslash sim\backslash mathrm\{Uniform\}(0,1)$ is a uniformly distributed variable on the unit interval, then $\backslash mu\; -\backslash beta\backslash log(-\backslash log(U))\backslash sim\backslash mathrm\{Gumbel\}(\backslash mu,\backslash beta)$.
• If $X\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash alpha\_X,\; \backslash beta)$ and $Y\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash alpha\_Y,\; \backslash beta)$ are independent, then $X-Y\; \backslash sim\; \backslash mathrm\{Logistic\}(\backslash alpha\_X-\backslash alpha\_Y,\backslash beta)\; \backslash ,$ (see Logistic distribution).
• Despite this, if $X,\; Y\; \backslash sim\; \backslash mathrm\{Gumbel\}(\backslash alpha,\; \backslash beta)$ are independent, then $X+Y\; \backslash nsim\; \backslash mathrm\{Logistic\}(2\; \backslash alpha,\backslash beta)$. This can easily be seen by noting that $\backslash mathbb\{E\}(X+Y)\; =\; 2\backslash alpha+2\backslash beta\backslash gamma\; \backslash neq\; 2\backslash alpha\; =\; \backslash mathbb\{E\}\backslash left(\backslash mathrm\{Logistic\}(2\; \backslash alpha,\backslash beta)\; \backslash right)$ (where $\backslash gamma$ is the Euler-Mascheroni constant). Instead, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.{{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}}

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

File:FitGumbelDistr.tif with confidence band of a cumulative Gumbel distribution to maximum one-day October rainfalls.{{Cite web|url=https://www.waterlog.info/cumfreq.htm|title=CumFreq, distribution fitting of probability, free calculator|website=www.waterlog.info}} ]]

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 {{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}} approaches the Gumbel distribution as the sample size increases.{{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}}

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

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

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

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,{{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 }} and also to describe droughts.{{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}}

Gumbel has also shown that the estimator {{frac|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 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 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{{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}} as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.{{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.

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

In analysis of algorithms, it appears, for example, in the study of the maximum carry propagation in base-$b$ addition algorithms.{{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}}

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

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

:$Q(p)=\backslash mu-\backslash beta\backslash ln(-\backslash ln(p)),$

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

File:Gumbel paper.JPG

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 $F$ :

: $-\backslash ln(-\backslash ln(F))\; =\; \backslash frac\{x-\backslash mu\}\backslash beta$

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

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".{{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}}

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

Equivalently, given any $x\_1,\; ...,\; x\_n\backslash in\; \backslash R$, we can sample from its Boltzmann distribution by

$$Pr(j\; =\; \backslash arg\backslash max\_i\; (g\_i\; +\; x\_i))\; =\; \backslash frac\{e^\{x\_j\}\}\{\backslash sum\_i\; e^\{x\_i\}\}$$Related equations include:{{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 }}

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

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

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