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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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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.
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 .
For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum (use the intermediate value theorem and Rolle's theorem to prove this by contradiction). In two and more dimensions, this argument fails.
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 ...
The extreme value theorem states that M is finite and f (c) = M for some c ∈ [a, b]. This can be proved by considering the set S = {s ∈ [a, b] : sup f ([s, b]) = M} . By definition of M, a ∈ S, and by its own definition, S is bounded by b. If c is the least upper bound of S, then it follows from continuity that f (c) = M.
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 theorem is typically interpreted as providing conditions for a parametric optimization problem to have continuous solutions with regard to the parameter. In this case, Θ {\displaystyle \Theta } is the parameter space, f ( x , θ ) {\displaystyle f(x,\theta )} is the function to be maximized, and C ( θ ) {\displaystyle C(\theta )} gives ...
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. [3] that a limit distribution needs to exist, which requires regularity conditions on the tail of the distribution. Despite this, the GEV distribution ...