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The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.
The Fisher–Tippett–Gnedenko theorem is a statement about the convergence of the limiting distribution , above. The study of conditions for convergence of to particular cases of the generalized extreme value distribution began with Mises (1936) [3] [5] [4] and was further developed by Gnedenko (1943).
In probability theory and statistics, the generalized extreme value (GEV) distribution [2] 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 ...
Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. It is widely used in many disciplines, such as structural engineering , finance , economics , earth sciences , traffic prediction, and geological engineering .
Extreme value theorem [ edit ] The extreme value theorem states that if a function f is defined on a closed interval [ a , b ] {\displaystyle [a,b]} (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c ∈ [ a , b ] {\displaystyle c\in [a,b]} with f ( c ) ≥ f ( x ) {\displaystyle ...
Cut-elimination theorem (proof theory) D. Dandelin's theorem ... Pickands–Balkema–de Haan theorem (extreme value theory) Pitman–Koopman–Darmois theorem
In conjunction with the extreme value theorem, it can be used to find the absolute maximum and minimum of a real-valued function defined on a closed and bounded interval. In conjunction with other information such as concavity, inflection points, and asymptotes, it can be used to sketch the graph of a function.
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 ).