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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value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930.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).
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] Note that a limit distribution needs to exist, which requires regularity conditions on the tail of the 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 .
The extreme value theorem states that if a function f is defined on a closed interval ... that satisfies a number of requirements, notably the triangle inequality.
Although the theorem is named after Michel Rolle, Rolle's 1691 proof covered only the case of polynomial functions. His proof did not use the methods of differential calculus, which at that point in his life he considered to be fallacious. The theorem was first proved by Cauchy in 1823 as a corollary of a proof of the mean value theorem. [1]
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.
By the extreme value theorem, it suffices that the likelihood function is continuous on a compact parameter space for the maximum likelihood estimator to exist. [7] While the continuity assumption is usually met, the compactness assumption about the parameter space is often not, as the bounds of the true parameter values might be unknown.