Gumbel distribution
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| Gumbel distribution | |
|---|---|
| Name | Gumbel distribution |
| Type | Continuous probability distribution |
| Support | (−∞, ∞) |
| Parameters | location μ, scale β>0 |
| Mean | μ + γβ |
| Variance | (π^2/6)β^2 |
Gumbel distribution The Gumbel distribution is a continuous probability distribution used to model the distribution of extreme values in samples drawn from many underlying distributions; it appears in extreme value theory connected to maxima and minima and finds application across hydrology, meteorology, materials science and actuarial science. It underpins models in flood risk, seismic hazard and structural reliability and is linked to seminal results by Fisher, Tippett and von Mises as well as applications by engineers at the United States Geological Survey and insurers such as Lloyd's of London.
Definition
A random variable X follows the Gumbel distribution with location parameter μ and scale parameter β>0 if its cumulative behavior of maxima arises under the Fisher–Tippett theorem alongside the Weibull and Fréchet domains; notable statisticians associated with this classification include Ronald Fisher, L. H. C. Tippett and Maurice Fréchet. The distribution is a limiting distribution for the maximum of a large collection of independent identically distributed random variables under conditions studied by Émile Borel and Richard von Mises and employed in publications from the Royal Society and Cambridge University Press. In applications, engineers at the U.S. Army Corps of Engineers, geoscientists at the United States Geological Survey and actuaries at the American Academy of Actuaries use parameterized Gumbel models for design storms, earthquake ground motion and extreme insurance losses.
Probability density and cumulative distribution
The cumulative distribution function (CDF) for the two-parameter Gumbel family is F(x)=exp(−exp(−(x−μ)/β)), a form that appears in textbooks by David Cox, D. R. Cox and Bradley Efron and in treatises by Harald Cramér and J. Neyman. The probability density function (PDF) is f(x)=(1/β) exp(−(z+e^{−z})) with z=(x−μ)/β, a representation used in monographs by A. Stuart, J. K. Ord and Carl Friedrich Gauss in likelihood derivations. The log-likelihood for independent observations leads to methods developed by Sir Ronald Fisher, Jerzy Neyman and Egon Pearson and is implemented in software from the R Project, SAS Institute and MathWorks.
Properties and moments
The Gumbel distribution has mean μ+γβ and variance (π^2/6)β^2 where γ denotes the Euler–Mascheroni constant introduced in work by Leonhard Euler and studied by Adrien-Marie Legendre and Srinivasa Ramanujan. Higher-order cumulants relate to values of the Riemann zeta function ζ(s) investigated by Bernhard Riemann and connected to asymptotic expansions in texts by G. H. Hardy. The distribution is in the max-domain of attraction of the Gumbel class identified in the Fisher–Tippett–Gnedenko theorem with theoretical development by Boris Gnedenko and contributions from Andrey Kolmogorov and Aleksandr Khinchin. Extreme value index estimators such as those of Hill and Pickands are used to assess tail behavior in empirical work from the International Association for Hydro-Environment Engineering and Research and the World Meteorological Organization.
Estimation and parameter inference
Parameter estimation for μ and β commonly employs maximum likelihood estimation (MLE) and method of moments, techniques formalized by Fisher and Neyman and applied in statistical packages from the R Foundation and Python Software Foundation. MLE for the Gumbel family is implemented via optimization algorithms originating with Davidon, Fletcher and Powell and numerical routines in libraries by John D. Cook and the Numerical Algorithms Group. Confidence intervals and hypothesis tests use asymptotic theory from C. R. Rao and Jerzy Neyman; practitioners in the insurance industry often augment inference with bootstrap resampling methods popularized by Bradley Efron and Robert Tibshirani and with Bayesian inference using priors discussed by Harold Jeffreys and Dennis Lindley.
Related distributions and applications
The Gumbel distribution is related to the generalized extreme value (GEV) distribution introduced by Samuel Coles and later used by the Intergovernmental Panel on Climate Change and the United Nations Office for Disaster Risk Reduction for climate extremes and disaster loss modeling. It connects to the exponential and double-exponential (Laplace) families and to logistic models used in epidemiology at the Centers for Disease Control and Prevention and in demography in United Nations reports. Applications span flood frequency analysis in reports by the U.S. Geological Survey, wind speed extremes in publications by the National Oceanic and Atmospheric Administration, structural reliability studies at the National Institute of Standards and Technology and fatigue life assessments in aerospace engineering by Airbus and Boeing.
History and naming
The distribution is named after Emil Julius Gumbel, a statistician and political activist whose work on extremes was contemporaneous with studies by Francis Ysidro Edgeworth and Karl Pearson and later integrated into the extreme value framework of Fisher and Tippett. Early applications appeared in hydrology and engineering literature from the United States Bureau of Reclamation and in meteorological studies at the Royal Meteorological Society; subsequent theoretical consolidation was influenced by the work of Maurice Fréchet, Boris Gnedenko and Andrey Kolmogorov. The naming reflects historical attribution practices found in the history of statistics alongside eponyms such as the Student's t distribution (William Sealy Gosset) and the Neyman–Pearson lemma.