“Extreme Value Distribution”的概念、定义、习题解题方法及翻译-数学辞典

单词

Extreme Value Distribution

释义

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)

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

, 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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