Gumbel distribution

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Short description: Particular case of the generalized extreme value distribution
Gumbel
Probability density function
Probability distribution function
Cumulative distribution function
Cumulative distribution function
Notation [math]\displaystyle{ \text{Gumbel}(\mu, \beta) }[/math]
Parameters [math]\displaystyle{ \mu, }[/math] location (real)
[math]\displaystyle{ \beta\gt 0, }[/math] scale (real)
Support [math]\displaystyle{ x\in\mathbb{R} }[/math]
PDF [math]\displaystyle{ \frac{1}{\beta}e^{-(z+e^{-z})} }[/math]
where [math]\displaystyle{ z=\frac{x-\mu}{\beta} }[/math]
CDF [math]\displaystyle{ e^{-e^{-(x-\mu)/\beta}} }[/math]
Mean [math]\displaystyle{ \mu + \beta\gamma }[/math]
where [math]\displaystyle{ \gamma }[/math] is the Euler–Mascheroni constant
Median [math]\displaystyle{ \mu - \beta\ln(\ln 2) }[/math]
Mode [math]\displaystyle{ \mu }[/math]
Variance [math]\displaystyle{ \frac{\pi^2}{6}\beta^2 }[/math]
Skewness [math]\displaystyle{ \frac{12\sqrt{6}\,\zeta(3)}{\pi^3} \approx 1.14 }[/math]
Ex. kurtosis [math]\displaystyle{ \frac{12}{5} }[/math]
Entropy [math]\displaystyle{ \ln(\beta)+\gamma+1 }[/math]
MGF [math]\displaystyle{ \Gamma(1-\beta t) e^{\mu t} }[/math]
CF [math]\displaystyle{ \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 variables has a logistic distribution.

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

Definitions

The cumulative distribution function of the Gumbel distribution is

[math]\displaystyle{ F(x;\mu,\beta) = e^{-e^{-(x-\mu)/\beta}}.\, }[/math]

Standard Gumbel distribution

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

[math]\displaystyle{ F(x) = e^{-e^{(-x)}}\, }[/math]

and probability density function

[math]\displaystyle{ f(x) = e^{-(x+e^{-x})}. }[/math]

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

The cumulants, for n>1, are given by

[math]\displaystyle{ \kappa_n = (n-1)! \zeta(n). }[/math]

Properties

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

[math]\displaystyle{ \operatorname{E}(X)=\mu+\gamma\beta }[/math],

where [math]\displaystyle{ \gamma }[/math] is the Euler-Mascheroni constant.

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

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

Related distributions

  • If [math]\displaystyle{ X }[/math] 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 [math]\displaystyle{ 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]\displaystyle{ g(y) = f(-y)/F(0) }[/math]: 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 [math]\displaystyle{ X \sim \mathrm{Gumbel}(\alpha_X, \beta) }[/math] and [math]\displaystyle{ Y \sim \mathrm{Gumbel}(\alpha_Y, \beta) }[/math] are independent, then [math]\displaystyle{ X-Y \sim \mathrm{Logistic}(\alpha_X-\alpha_Y,\beta) \, }[/math] (see Logistic distribution).
  • If [math]\displaystyle{ X, Y \sim \mathrm{Gumbel}(\alpha, \beta) }[/math] are independent, then [math]\displaystyle{ X+Y \nsim \mathrm{Logistic}(2 \alpha,\beta) }[/math]. Note that [math]\displaystyle{ 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.

Occurrence and applications

Distribution fitting with confidence band of a cumulative Gumbel distribution to maximum one-day October rainfalls.[5]

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

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

[math]\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 }[/math],

and, for large [math]\displaystyle{ N }[/math], the right-hand-side converges to [math]\displaystyle{ 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,[3] and also to describe droughts.[8]

Gumbel has also shown that the estimatorr(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[9] as well as the trend-adjusted sizes of maximal prime gaps and maximal gaps between prime constellations.[10]

Gumbel reparametrization 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 "reparametrization tricks".[11]

In detail, let [math]\displaystyle{ (\pi_1, ..., \pi_n) }[/math] be nonnegative, and not all zero, and let [math]\displaystyle{ g_1,... , g_n }[/math] be independent samples of Gumbel(0, 1), then by routine integration,[math]\displaystyle{ Pr(j = \arg\max_i (g_i + \log\pi_i)) = \frac{\pi_j}{\sum_i \pi_i} }[/math]That is, [math]\displaystyle{ \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]\displaystyle{ x_1, ..., x_n\in \R }[/math], we can sample from its Boltzmann distribution by

[math]\displaystyle{ Pr(j = \arg\max_i (g_i + x_i)) = \frac{e^{x_j}}{\sum_i e^{x_i}} }[/math]Related equations include:[12]

  • If [math]\displaystyle{ x\sim Exp(\lambda) }[/math], then [math]\displaystyle{ (-\ln x - \gamma)\sim \text{Gumbel}(-\gamma + \ln\lambda, 1) }[/math].
  • [math]\displaystyle{ \arg\max_i (g_i + \log\pi_i) \sim \text{Categorical}\left(\frac{\pi_j}{\sum_i \pi_i}\right)_j }[/math].
  • [math]\displaystyle{ \max_i (g_i + \log\pi_i) \sim \text{Gumbel}\left(-\gamma + \log\left(\sum_i \pi_i \right), 1\right) }[/math]. That is, the Gumbel distribution is a max-stable distribution family.
  • [math]\displaystyle{ \mathbb E[\max_i (g_i + \beta x_i)] = \log \left(\sum_i e^{\beta x_i}\right) }[/math].

Computational methods

Probability paper

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]\displaystyle{ F }[/math] :

[math]\displaystyle{ -\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]\displaystyle{ F }[/math] on the horizontal axis of the paper and the [math]\displaystyle{ x }[/math]-variable on the vertical axis, the distribution is represented by a straight line with a slope 1[math]\displaystyle{ /\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.

Generating Gumbel variates

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

[math]\displaystyle{ Q(p)=\mu-\beta\ln(-\ln(p)), }[/math]

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

See also

References

  1. Gumbel, E.J. (1935), "Les valeurs extrêmes des distributions statistiques", Annales de l'Institut Henri Poincaré 5 (2): 115–158, http://archive.numdam.org/article/AIHP_1935__5_2_115_0.pdf 
  2. Gumbel E.J. (1941). "The return period of flood flows". The Annals of Mathematical Statistics, 12, 163–190.
  3. 3.0 3.1 Oosterbaan, R.J. (1994). "Chapter 6 Frequency and Regression Analysis". in Ritzema, H.P.. Drainage Principles and Applications, Publication 16. Wageningen, The Netherlands: International Institute for Land Reclamation and Improvement (ILRI). pp. 175–224. ISBN 90-70754-33-9. http://www.waterlog.info/pdf/freqtxt.pdf. 
  4. Willemse, W.J.; Kaas, R. (2007). "Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz' law of mortality". Insurance: Mathematics and Economics 40 (3): 468. doi:10.1016/j.insmatheco.2006.07.003. https://www.dnb.nl/binaries/Working%20Paper%20135-2007_tcm46-146792.pdf. 
  5. CumFreq, software for probability distribution fitting
  6. user49229, Gumbel distribution and exponential distribution
  7. Gumbel, E.J. (1954). Statistical theory of extreme values and some practical applications. Applied Mathematics Series. 33 (1st ed.). U.S. Department of Commerce, National Bureau of Standards. https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/PB175818.xhtml. 
  8. Burke, Eleanor J.; Perry, Richard H.J.; Brown, Simon J. (2010). "An extreme value analysis of UK drought and projections of change in the future". Journal of Hydrology 388 (1–2): 131–143. doi:10.1016/j.jhydrol.2010.04.035. Bibcode2010JHyd..388..131B. 
  9. Erdös, Paul; Lehner, Joseph (1941). "The distribution of the number of summands in the partitions of a positive integer". Duke Mathematical Journal 8 (2): 335. doi:10.1215/S0012-7094-41-00826-8. 
  10. Kourbatov, A. (2013). "Maximal gaps between prime k-tuples: a statistical approach". Journal of Integer Sequences 16. Bibcode2013arXiv1301.2242K.  Article 13.5.2.
  11. Jang, Eric; Gu, Shixiang; Poole, Ben (April 2017). "Categorical Reparametrization with Gumble-Softmax". International Conference on Learning Representations (ICLR) 2017. https://pure.mpg.de/pubman/faces/ViewItemOverviewPage.jsp?itemId=item_2564872. 
  12. Balog, Matej; Tripuraneni, Nilesh; Ghahramani, Zoubin; Weller, Adrian (2017-07-17). "Lost Relatives of the Gumbel Trick" (in en). International Conference on Machine Learning (PMLR): 371–379. https://proceedings.mlr.press/v70/balog17a.html. 

External links