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Extremwertverteilung 極値分布 Loi d'extremum généralisée Distribuzione generalizzata dei valori estremi Generalized extreme value distribution
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極値分布（きょくちぶんぷ、英: extreme value distribution）とは、確率論および統計学において、ある累積分布関数にしたがって生じた大きさ n の標本 X1,X2, …, Xn のうち、x 以上 (あるいは以下) となるものの個数がどのように分布するかを表す、連続確率分布モデルである。特に最大値や最小値などが漸近的に従う分布であり、河川の氾濫、最大風速、最大降雨量、金融におけるリスク等の分布に適用される。 In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables. Die verallgemeinerte Extremwertverteilung ist eine stetige Wahrscheinlichkeitsverteilung. Sie spielt eine herausragende Rolle in der Extremwerttheorie, da sie alle möglichen asymptotischen Verteilungen des Maximums einer einfachen Zufallsstichprobe in einer Darstellung zusammenfasst.Die verallgemeinerte Extremwertverteilung fasst die Gumbel-Verteilung, die Fréchet-Verteilung und die Weibull-Verteilung zusammen. In teoria della probabilità la distribuzione generalizzata dei valori estremi (dall'inglese generalized extreme value distribution, in sigla GEV), o distribuzione di Fisher-Tippett, è una famiglia di distribuzioni di probabilità che raccoglie la distribuzione di Fréchet, la distribuzione di Weibull e la distribuzione di Gumbel (come caso al limite). Questa famiglia è comune nella teoria dei valori estremi, dove descrive il limite dei massimi in una successione di variabili aleatorie indipendenti, secondo il . En probabilité et statistique, la loi d'extrémum généralisée est une famille de lois de probabilité continues qui servent à représenter des phénomènes de valeurs extrêmes (minimum ou maximum). Elle comprend la loi de Gumbel, la loi de Fréchet et la loi de Weibull, respectivement lois d'extrémum de type I, II et III. Le théorème de Fisher-Tippett-Gnedenko établit que la loi d'extremum généralisée est la distribution limite du maximum (adéquatement normalisé) d'une série de variables aléatoires indépendantes de même distribution (iid).
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In teoria della probabilità la distribuzione generalizzata dei valori estremi (dall'inglese generalized extreme value distribution, in sigla GEV), o distribuzione di Fisher-Tippett, è una famiglia di distribuzioni di probabilità che raccoglie la distribuzione di Fréchet, la distribuzione di Weibull e la distribuzione di Gumbel (come caso al limite). Questa famiglia è comune nella teoria dei valori estremi, dove descrive il limite dei massimi in una successione di variabili aleatorie indipendenti, secondo il . Il secondo nome con cui è conosciuta deriva dagli statistici britannici Fisher e Tippett. En probabilité et statistique, la loi d'extrémum généralisée est une famille de lois de probabilité continues qui servent à représenter des phénomènes de valeurs extrêmes (minimum ou maximum). Elle comprend la loi de Gumbel, la loi de Fréchet et la loi de Weibull, respectivement lois d'extrémum de type I, II et III. Le théorème de Fisher-Tippett-Gnedenko établit que la loi d'extremum généralisée est la distribution limite du maximum (adéquatement normalisé) d'une série de variables aléatoires indépendantes de même distribution (iid). La loi d'extrémum généralisée est connue sous le nom de loi de Fisher-Tippett, d'après Ronald Fisher et L. H. C. Tippett qui ont étudié les trois formes fonctionnelles ci-dessous. Parfois, ce nom signifie plus particulièrement le cas de la loi de Gumbel. 極値分布（きょくちぶんぷ、英: extreme value distribution）とは、確率論および統計学において、ある累積分布関数にしたがって生じた大きさ n の標本 X1,X2, …, Xn のうち、x 以上 (あるいは以下) となるものの個数がどのように分布するかを表す、連続確率分布モデルである。特に最大値や最小値などが漸近的に従う分布であり、河川の氾濫、最大風速、最大降雨量、金融におけるリスク等の分布に適用される。 Die verallgemeinerte Extremwertverteilung ist eine stetige Wahrscheinlichkeitsverteilung. Sie spielt eine herausragende Rolle in der Extremwerttheorie, da sie alle möglichen asymptotischen Verteilungen des Maximums einer einfachen Zufallsstichprobe in einer Darstellung zusammenfasst.Die verallgemeinerte Extremwertverteilung fasst die Gumbel-Verteilung, die Fréchet-Verteilung und die Weibull-Verteilung zusammen. In probability theory and statistics, the generalized extreme value (GEV) distribution is a family of continuous probability distributions developed within extreme value theory to combine the Gumbel, Fréchet and Weibull families also known as type I, II and III extreme value distributions. By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution is often used as an approximation to model the maxima of long (finite) sequences of random variables. In some fields of application the generalized extreme value distribution is known as the Fisher–Tippett distribution, named after Ronald Fisher and L. H. C. Tippett who recognised three different forms outlined below. However usage of this name is sometimes restricted to mean the special case of the Gumbel distribution. The origin of the common functional form for all 3 distributions dates back to at least Jenkinson, A. F. (1955), though allegedly it could also have been given by von Mises, R. (1936).
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μ ∈ R — location, ξ ∈ R — shape. σ > 0 — scale,
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