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Fisher-Tippett Distribution

\begin{figure}\begin{center}\BoxedEPSF{ExtremeValueDistribution.epsf scaled 650}\end{center}\end{figure}

Also called the Extreme Value Distribution and Log-Weibull Distribution. It is the limiting distribution for the smallest or largest values in a large sample drawn from a variety of distributions.

$\displaystyle P(x)$ $\textstyle =$ $\displaystyle {e^{{(a-x)/b}-e^{(a-x)/b}}\over b}$ (1)
$\displaystyle D(x)$ $\textstyle =$ $\displaystyle e^{-e^{(a-x)/b}}.$ (2)

These can be computed directly be defining
$\displaystyle z$ $\textstyle \equiv$ $\displaystyle \mathop{\rm exp}\nolimits \left({a-x\over b}\right)$ (3)
$\displaystyle x$ $\textstyle =$ $\displaystyle a-b\ln z$ (4)
$\displaystyle dz$ $\textstyle =$ $\displaystyle -{1\over b}\mathop{\rm exp}\nolimits \left({a-x\over b}\right)\,dx.$ (5)

Then the Moments are
$\displaystyle \mu_n$ $\textstyle \equiv$ $\displaystyle \int_{-\infty}^\infty x^nP(x)\,dx$  
  $\textstyle =$ $\displaystyle {1\over b}\int_{-\infty}^\infty x^n\mathop{\rm exp}\nolimits \left({a\over x-b}\right)\mathop{\rm exp}\nolimits [-e^{(a-x)/b}]\,dx$  
  $\textstyle =$ $\displaystyle -\int_\infty^0 (a-b\ln z)^n e^{-z}\,dz$  
  $\textstyle =$ $\displaystyle \int_0^\infty (a-b\ln z)^n e^{-z}\,dz$  
  $\textstyle =$ $\displaystyle \sum_{k=0}^n {n\choose k} (-1)^k a^{n-k} b^k \int_0^\infty (\ln z)^k e^{-z}\,dz$  
  $\textstyle =$ $\displaystyle \sum_{k=0}^n {n\choose k} a^{n-k} b^k I(k),$ (6)

where $I(k)$ are Euler-Mascheroni Integrals. Plugging in the Euler-Mascheroni Integrals $I(k)$ gives

$\displaystyle \mu_0$ $\textstyle =$ $\displaystyle 1$ (7)
$\displaystyle \mu_1$ $\textstyle =$ $\displaystyle a+b\gamma$ (8)
$\displaystyle \mu_2$ $\textstyle =$ $\displaystyle a^2+2ab\gamma+b^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)$ (9)
$\displaystyle \mu_3$ $\textstyle =$ $\displaystyle a^3+3a^2b\gamma+3ab^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+b^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$ (10)
$\displaystyle \mu_4$ $\textstyle =$ $\displaystyle a^4+4a^3b\gamma+6a^2b^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+4ab^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$  
  $\textstyle \phantom{=}$ $\displaystyle +b^4[\gamma^4+\gamma^2\pi^2+{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3)],$ (11)

where $\gamma$ is the Euler-Mascheroni Constant and $\zeta(3)$ is Apéry's Constant. The Mean, Variance, Skewness, and Kurtosis are therefore

$\displaystyle \mu$ $\textstyle =$ $\displaystyle a+b\gamma$ (12)
$\displaystyle \sigma^2$ $\textstyle =$ $\displaystyle \mu_2-{\mu_1}^2={\textstyle{1\over 6}} \pi^2 b^2$ (13)
$\displaystyle \gamma_1$ $\textstyle =$ $\displaystyle {\mu_3\over\sigma^3}$  
  $\textstyle =$ $\displaystyle {6\sqrt{6}\over b^3\pi^3}\{a^3+3a^2b\gamma+3ab^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+b^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]\}$ (14)
$\displaystyle \gamma_2$ $\textstyle =$ $\displaystyle {\mu_4\over\sigma^4}-3$  
  $\textstyle =$ $\displaystyle {36\over b^4\pi^4} \{a^4+4a^3b\gamma+a^2b^2(6\gamma^2+\pi^2)+4ab^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$  
  $\textstyle \phantom{=}$ $\displaystyle +b^4[\gamma^4+\gamma^2\pi^2+{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3)]\}.$ (15)

The Characteristic Function is
\phi(t)=\Gamma(1-i\beta t)e^{i\alpha t},
\end{displaymath} (16)

where $\Gamma(z)$ is the Gamma Function. The special case of the Fisher-Tippett distribution with $a=0$, $b=1$ is called Gumbel's Distribution.

See also Euler-Mascheroni Integrals, Gumbel's Distribution

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© 1996-9 Eric W. Weisstein