*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) |

(2) | |||

(3) |

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

(4) |

(5) |

(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.

© 1996-9

1999-05-25