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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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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).
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 .
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 ).
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 ...
The extreme value theorem states that if a function f is defined on a closed interval [,] ... An extreme example: if a set X is given the discrete topology ...
The Weierstrass approximation theorem, of which one well known generalization is the Stone–Weierstrass theorem; The Bolzano–Weierstrass theorem, which ensures compactness of closed and bounded sets in R n; The Weierstrass extreme value theorem, which states that a continuous function on a closed and bounded set obtains its extreme values
Closed graph theorem (functional analysis) Extreme value theorem ; Fixed-point theorems in infinite-dimensional spaces; Hairy ball theorem (algebraic topology) Hahn–Mazurkiewicz theorem (continuum theory) Heine–Borel theorem (real analysis) Heine–Cantor theorem (metric geometry) Jordan curve theorem