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

$$P(M_n<x)=\cases{ 0 & if $x<0$\cr x^n & if $0\leq x\leq 1$\cr 1 & if $x>1$,\cr}$$ (1)

and the Mean and Variance are

$$\mu = {n\over n+1}$$ (2)

$$\sigma^2 = {n\over(n+1)^2(n+2)}.$$ (3)

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

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

where $\Phi(x)$ is the Normal Distribution Function. The probability distribution of $M_n$ is then

$$P(M_n<x)=[F(x)]^n={n\over\sqrt{2\pi}}\int_{-\infty}^x [F(t)]^{n-1}e^{-t^2/2}\,dt.$$ (5)

The Mean $\mu(n)$ and Variance $\sigma^2(n)$ are expressible in closed form for small $n$,

$$\mu(1) = 0$$ (6)

$$\mu(2) = {1\over \sqrt{\pi}}$$ (7)

$$\mu(3) = {3\over 2\sqrt{\pi}}$$ (8)

$$\mu(4) = {3\over 2\sqrt{\pi}}\left[{1+{2\over\pi}\sin^{-1}({\textstyle{1\over 3}})}\right]$$ (9)

$$\mu(5) = {5\over 4\sqrt{\pi}}\left[{1+{6\over\pi}\sin^{-1}({\textstyle{1\over 3}})}\right]$$ (10)

and

$$\sigma^2(1) = 1$$ (11)

$$\sigma^2(2) = 1-{1\over\pi}$$ (12)

$$\sigma^2(3) = {4\pi-9+2\sqrt{3}\over 4\pi}$$ (13)

$$\sigma^2(4) = 1+{\sqrt{3}\over\pi}-[\mu(4)]^2$$ (14)

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

No exact expression is known for $\mu(6)$ or $\sigma^2(6)$, but there is an equation connecting them

$$[\mu(6)]^2+\sigma^2(6)=1+{5\sqrt{3}\over 4\pi}+{15\sqrt{3}\over 2\pi^2}\sin^{-1}({\textstyle{1\over 4}}).$$ (16)

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

$$P(y) = \mathop{\rm exp}\nolimits (-e^{-y})$$ (17)

$$P(y) = \left\{\begin{array}{ll} 0 & \mbox{if $y\leq 0$}\\ \mathop{\rm exp}\nolimits (-y^{-a}) & \mbox{if $y>0$}\end{array}\right.$$ (18)

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

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

See also Fisher-Tippett Distribution, Order Statistic

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.