Extreme Value Distribution
N.B. A detailed on-line essay by S. Finchwas the starting point for this entry.
Let 
denote the ``extreme'' (i.e., largest) Order Statistic 
for a distribution of 
elements
taken from a continuous Uniform Distribution. Then the distribution of the is
 | (1) |
 |  |  | (2) |
 | |  | (3) |
If are taken from a Standard Normal Distribution, then its cumulative distribution is
 | (4) |
is the Normal Distribution Function. The probability distribution of is then | (5) |
and Variance 
are expressible in closed form for small , | |  | (6) |
 | |  | (7) |
 | |  | (8) |
 | |  | (9) |
 | |  | (10) |
and
 | |  | (11) |
 | |  | (12) |
 | |  | (13) |
 | |  | (14) |
 | |  | (15) |
No exact expression is known for
or 
, but there is an equation connecting them | (16) |
An analog to the Central Limit Theorem states that the asymptotic normalized distribution of satisfies one of thethree distributions
 | |  | (17) |
| |  | (18) |
| |  | (19) |
also known as Gumbel, Fréchet, and Weibull Distributions,respectively.See also Fisher-Tippett Distribution, Order Statistic
References
Balakrishnan, N. and Cohen, A. C. Order Statistics and Inference. New York: Academic Press, 1991. David, H. A. Order Statistics, 2nd ed. New York: Wiley, 1981. Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/extval/extval.html Gibbons, J. D. and Chakraborti, S. Nonparametric Statistical Inference, 3rd rev. ext. ed. New York: Dekker, 1992.
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