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Extreme Value Distribution

N.B. A detailed on-line essay by S. Finch was the starting point for this entry.

Let $M_n$ denote the ``extreme'' (i.e., largest) Order Statistic $X^{\left\langle{n}\right\rangle{}}$ for a distribution of $n$ elements $X_i$ taken from a continuous Uniform Distribution. Then the distribution of the $M_n$ is

0 & if $x<0$\cr
x^n & if $0\leq x\leq 1$\cr
1 & if $x>1$,\cr}
\end{displaymath} (1)

and the Mean and Variance are
$\displaystyle \mu$ $\textstyle =$ $\displaystyle {n\over n+1}$ (2)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle {n\over(n+1)^2(n+2)}.$ (3)

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

F(x)={1\over\sqrt{2\pi}}\int_{-\infty}^x e^{-t^2/2}\,dt = {\textstyle{1\over 2}}+\Phi(x),
\end{displaymath} (4)

where $\Phi(x)$ is the Normal Distribution Function. The probability distribution of $M_n$ is then
P(M_n<x)=[F(x)]^n={n\over\sqrt{2\pi}}\int_{-\infty}^x [F(t)]^{n-1}e^{-t^2/2}\,dt.
\end{displaymath} (5)

The Mean $\mu(n)$ and Variance $\sigma^2(n)$ are expressible in closed form for small $n$,
$\displaystyle \mu(1)$ $\textstyle =$ $\displaystyle 0$ (6)
$\displaystyle \mu(2)$ $\textstyle =$ $\displaystyle {1\over \sqrt{\pi}}$ (7)
$\displaystyle \mu(3)$ $\textstyle =$ $\displaystyle {3\over 2\sqrt{\pi}}$ (8)
$\displaystyle \mu(4)$ $\textstyle =$ $\displaystyle {3\over 2\sqrt{\pi}}\left[{1+{2\over\pi}\sin^{-1}({\textstyle{1\over 3}})}\right]$ (9)
$\displaystyle \mu(5)$ $\textstyle =$ $\displaystyle {5\over 4\sqrt{\pi}}\left[{1+{6\over\pi}\sin^{-1}({\textstyle{1\over 3}})}\right]$ (10)

$\displaystyle \sigma^2(1)$ $\textstyle =$ $\displaystyle 1$ (11)
$\displaystyle \sigma^2(2)$ $\textstyle =$ $\displaystyle 1-{1\over\pi}$ (12)
$\displaystyle \sigma^2(3)$ $\textstyle =$ $\displaystyle {4\pi-9+2\sqrt{3}\over 4\pi}$ (13)
$\displaystyle \sigma^2(4)$ $\textstyle =$ $\displaystyle 1+{\sqrt{3}\over\pi}-[\mu(4)]^2$ (14)
$\displaystyle \sigma^2(5)$ $\textstyle =$ $\displaystyle 1+{5\sqrt{3}\over 4\pi}+{5\sqrt{3}\over 2\pi^2}\sin^{-1}({\textstyle{1\over 4}})-[\mu(5)]^2.$ (15)

No exact expression is known for $\mu(6)$ or $\sigma^2(6)$, but there is an equation connecting them
\begin{displaymath}[\mu(6)]^2+\sigma^2(6)=1+{5\sqrt{3}\over 4\pi}+{15\sqrt{3}\over 2\pi^2}\sin^{-1}({\textstyle{1\over 4}}).
\end{displaymath} (16)

An analog to the Central Limit Theorem states that the asymptotic normalized distribution of $M_n$ satisfies one of the three distributions

$\displaystyle P(y)$ $\textstyle =$ $\displaystyle \mathop{\rm exp}\nolimits (-e^{-y})$ (17)
$\displaystyle P(y)$ $\textstyle =$ $\displaystyle \left\{\begin{array}{ll} 0 & \mbox{if $y\leq 0$}\\  \mathop{\rm exp}\nolimits (-y^{-a}) & \mbox{if $y>0$}\end{array}\right.$ (18)
$\displaystyle P(y)$ $\textstyle =$ $\displaystyle \left\{\begin{array}{ll} \mathop{\rm exp}\nolimits [-(-y)^a] & \mbox{if $y\leq 0$}\\  1 & \mbox{if $y>0$,}\end{array}\right.$ (19)

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

See also Fisher-Tippett Distribution, Order Statistic


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.''

Gibbons, J. D. and Chakraborti, S. Nonparametric Statistical Inference, 3rd rev. ext. ed. New York: Dekker, 1992.

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© 1996-9 Eric W. Weisstein