## Extreme Value Distribution

N.B. A detailed on-line essay by S. Finch was 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)

and the Mean and Variance are
 (2) (3)

If are taken from a Standard Normal Distribution, then its cumulative distribution is

 (4)

where is the Normal Distribution Function. The probability distribution of is then
 (5)

The Mean 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 the three distributions

 (17) (18) (19)

also known as Gumbel, Fréchet, and Weibull Distributions, respectively.

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.