Fisher-Tippett Distribution

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.

$$P(x) = {e^{{(a-x)/b}-e^{(a-x)/b}}\over b}$$ (1)

$$D(x) = e^{-e^{(a-x)/b}}.$$ (2)

These can be computed directly be defining

$$z \equiv \mathop{\rm exp}\nolimits \left({a-x\over b}\right)$$ (3)

$$x = a-b\ln z$$ (4)

$$dz = -{1\over b}\mathop{\rm exp}\nolimits \left({a-x\over b}\right)\,dx.$$ (5)

Then the Moments are

$$\mu_n \equiv \int_{-\infty}^\infty x^nP(x)\,dx$$

$$= {1\over b}\int_{-\infty}^\infty x^n\mathop{\rm exp}\nolimits \left({a\over x-b}\right)\mathop{\rm exp}\nolimits [-e^{(a-x)/b}]\,dx$$

$$= -\int_\infty^0 (a-b\ln z)^n e^{-z}\,dz$$

$$= \int_0^\infty (a-b\ln z)^n e^{-z}\,dz$$

$$= \sum_{k=0}^n {n\choose k} (-1)^k a^{n-k} b^k \int_0^\infty (\ln z)^k e^{-z}\,dz$$

$$= \sum_{k=0}^n {n\choose k} a^{n-k} b^k I(k),$$ (6)

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

$$\mu_0 = 1$$ (7)

$$\mu_1 = a+b\gamma$$ (8)

$$\mu_2 = a^2+2ab\gamma+b^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)$$ (9)

$$\mu_3 = a^3+3a^2b\gamma+3ab^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+b^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$$ (10)

$$\mu_4 = a^4+4a^3b\gamma+6a^2b^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+4ab^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$$

$$\phantom{=} +b^4[\gamma^4+\gamma^2\pi^2+{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3)],$$ (11)

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

$$\mu = a+b\gamma$$ (12)

$$\sigma^2 = \mu_2-{\mu_1}^2={\textstyle{1\over 6}} \pi^2 b^2$$ (13)

$$\gamma_1 = {\mu_3\over\sigma^3}$$

$$= {6\sqrt{6}\over b^3\pi^3}\{a^3+3a^2b\gamma+3ab^2(\gamma^2+{\textstyle{1\over 6}}\pi^2)+b^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]\}$$ (14)

$$\gamma_2 = {\mu_4\over\sigma^4}-3$$

$$= {36\over b^4\pi^4} \{a^4+4a^3b\gamma+a^2b^2(6\gamma^2+\pi^2)+4ab^3[\gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)]$$

$$\phantom{=} +b^4[\gamma^4+\gamma^2\pi^2+{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3)]\}.$$ (15)

The Characteristic Function is

$$\phi(t)=\Gamma(1-i\beta t)e^{i\alpha t},$$ (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