PLAYGROUND

ECUACIONES DISTRIBUCIÓN GENERALIZED EXTREME VALUE

Definición de distribución

X∼GeneralizedExtremeValue(ξ,μ,σ)X\sim\mathrm{GeneralizedExtremeValue}\left(\xi,\mu,\sigma\right)

Dominio de distribución

if ξ>0: x∈(z(x),∞),if ξ=0: x∈(−∞,∞),if ξ<0: x∈(−∞,z(x))\text{if }\xi>0:\ x\in\left(z(x),\infty\right),\quad \text{if }\xi=0:\ x\in\left(-\infty,\infty\right),\quad \text{if }\xi<0:\ x\in\left(-\infty,z(x)\right)

Dominio y restricciones de parámetros

ξ∈R,μ∈R,σ∈R+\xi\in\mathbb{R},\mu\in\mathbb{R},\sigma\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)={exp⁡(−exp⁡(−z(x)))if  ξ=0exp⁡(−(1+ξz(x))−1/ξ)if  ξ≠0F_{X}\left(x\right)=\left\{\begin{array}{cl} \exp\Bigl(-\exp(-z(x))\Bigr) & \text{if } ~ \xi=0 \\ \exp\Bigl(-(1+\xi z(x))^{-1/\xi}\Bigr) & \text{if } ~ \xi \neq 0\end{array} \right.

Función de densidad de probabilidad

fX(x)={1σexp⁡(−z(x))exp⁡(−exp⁡(−z(x)))if  ξ=01σ(1+ξz(x))−(1+1/ξ)exp⁡(−(1+ξz(x))−1/ξ)if  ξ≠0f_{X}\left(x\right)=\left\{\begin{array}{cl}\frac{1}{\sigma}\exp(-z(x)) \exp\Bigl(-\exp(-z(x))\Bigr) & \text{if } ~ \xi=0 \\ \frac{1}{\sigma}\Bigl(1+\xi z(x)\Bigr)^{-(1+1/\xi)} \exp\Bigl(-(1+\xi z(x))^{-1/\xi}\Bigr) & \text{if } ~ \xi \neq 0\end{array} \right.

Función de punto percentil

FX−1(u)={μ−σln⁡(−ln⁡(u))if  ξ=0μ+σξ((−ln⁡(u) )−ξ−1)if  ξ≠0F^{-1}_{X}\left(u\right)=\left\{\begin{array}{cl} \mu-\sigma\ln\left(-\ln\left(u\right)\right) & \text{if } ~ \xi=0 \\ \mu+\frac{}{}\frac{\sigma}{\xi}\left(\left(-\ln(u)\,\right)^{-\xi}-1\right) & \text{if } ~ \xi \neq 0\\ \end{array} \right.

Momentos paramétricos no centrados

μk′=E[Xk]=∫−∞∞xkfX(x)dx=Γ(1−kξ)\mu'_{k}=E[X^k]=\int_{-\infty}^{\infty}x^{k}f_{X}\left(x\right)dx=\Gamma(1-k\xi)

Media paramétrica

Mean(X)={μ+σ(μ1′−1)/ξif ξ≠0,ξ<1μ+σ γif ξ=0\mathrm{Mean}(X)=\left\{\begin{array}{cl}\mu+\sigma(\mu'_{1}-1)/\xi & \text{if}\ \xi\neq 0,\xi<1\\ \mu+\sigma\,\gamma & \text{if}\ \xi=0\end{array} \right.

Varianza paramétrica

Variance(X)={σ2 (μ2′−μ1′2)/ξ2if ξ≠0,ξ<12σ2 π26if ξ=0\mathrm{Variance}(X)=\left\{\begin{array}{cl}\sigma^2\,(\mu'_{2}-\mu'^{2}_{1})/\xi^2 & \text{if}\ \xi\neq0,\xi<\frac12\\ \sigma^2\,\frac{\pi^2}{6} & \text{if}\ \xi=0\end{array} \right.

Coeficiente de asimetría paramétrico

Skewness(X)={sign(ξ)μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5if ξ≠0,ξ<13126 ζ(3)π3if ξ=0\mathrm{Skewness}(X)=\left\{\begin{array}{cl}\text{sign}(\xi)\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}} & \text{if}\ \xi\neq0,\xi<\frac{1}{3} \\ \frac{12 \sqrt{6}\,\zeta(3)}{\pi^3} & \text{if}\ \xi=0\end{array} \right.

Curtosis paramétrica

Kurtosis(X)={3+μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2if ξ≠0,ξ<143+125if ξ=0\mathrm{Kurtosis}(X)=\left\{\begin{array}{cl}3+\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}} & \text{if}\ \xi\neq0,\xi<\frac{1}{4}\\ 3+\frac{12}{5} & \text{if}\ \xi=0\end{array} \right.

Mediana paramétrica

Median(X)={μ+σ(ln⁡2)−ξ−1ξif  ξ≠0μ−σln⁡ln⁡2if ξ=0\mathrm{Median}(X)=\left\{\begin{array}{cl}\mu+\sigma\frac{(\ln2)^{-\xi}-1}{\xi} & \text{if }\ \xi\neq0\\ \mu-\sigma \ln\ln2 & \text{if}\ \xi=0\end{array} \right.

Moda paramétrica

Mode(X)={μ+σ(1+ξ)−ξ−1ξif  ξ≠0μif  ξ=0\mathrm{Mode}(X)=\left\{\begin{array}{cl}\mu+\sigma\frac{(1+\xi)^{-\xi}-1}{\xi} & \text{if }\ \xi\neq0\\ \mu & \text{if }\ \xi=0\end{array} \right.

Información y definiciones adicionales

μ:Location parameter\mu:\text{Location parameter}
σ:Scale parameter\sigma:\text{Scale parameter}
z(x)=(x−μ)/σz\left(x\right)=\left(x-\mu\right)/\sigma
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649