Extreme value theory is the branch of probability concerned with the limiting behavior of unusually large or small observations. Its classical starting point is the maximum
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(1)
|
of independent and identically distributed random variates with common distribution function 
. For finite 
,
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(2)
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This distribution usually becomes degenerate as 
tends to infinity, so constants 
and 
are sought for which
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at every continuity point of a nondegenerate distribution function 
.
The Fisher-Tippett-Gnedenko theorem states that, if such a limit exists, then after a choice of location and scale it has the generalized extreme value form
![G_xi(x)=exp[-(1+xix)^(-1/xi)]](/images/equations/ExtremeValueTheory/NumberedEquation4.svg) |
(4)
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on the set where 
.
The limiting case 
is interpreted as
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(5)
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The cases 
,
,
and 
correspond to the Gumbel distribution, the
Fréchet type, and the Weibull
type, respectively. Thus, unlike the central
limit theorem, the extreme value theorem has three types of nondegenerate limits
rather than a single type (Fisher and Tippett 1928, Gnedenko 1943).
A distribution 
for which suitable 
and 
give the limit 
is said to belong to the maximum domain of attraction of 
. Roughly, exponentially decreasing tails, including those
of the normal distribution and exponential
distribution, give the Gumbel type. Power-law tails such as that of the Pareto
distribution give the Fréchet type,
while distributions having a finite upper endpoint and an appropriate power-law approach
to that endpoint give the Weibull type. The parameter 
is commonly called the extreme value index or tail index.
The same normalization also describes the upper order statistics. If 
is the 
th
largest observation and the normalized maximum converges to the standard Gumbel distribution,
then for fixed 
,
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This formula expresses the limiting number of observations above a high normalized level in terms of a Poisson distribution.
Results for minima follow by applying the theory to 
. Extensions of extreme value theory treat dependent observations,
exceedances over high thresholds, and point-process limits (Leadbetter et al. 1983,
Resnick 1987, de Haan and Ferreira 2006).
