PLAYGROUND

ECUACIONES DISTRIBUCIÓN LOGISTICA GENERALIZADA

Definición de distribución

X∼GeneralizedLogistic(c,Loc,Sc)X\sim\mathrm{GeneralizedLogistic}\left(c,\text{Loc},\text{Sc}\right)

Dominio de distribución

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Dominio y restricciones de parámetros

c∈R+,Loc∈R,Sc∈R+c\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=1(1+exp⁡(−z(x)))cF_{X}\left(x\right)=\frac{1}{\left(1+\exp\left(-z(x)\right)\right)^{c}}

Función de densidad de probabilidad

fX(x)=cexp⁡(−z(x))Sc(1+exp⁡(−z(x)))c+1f_{X}\left(x\right)=\frac{c\exp\left(-z(x)\right)}{\text{Sc}\left(1+\exp\left(-z(x)\right)\right)^{c+1}}

Función de punto percentil

FX−1(u)=Loc−Scln⁡(u−1/c−1)F^{-1}_{X}\left(u\right)=\text{Loc}-\text{Sc}\ln\left(u^{-1/c}-1\right)

Momentos paramétricos no centrados

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

Media paramétrica

Mean(X)=Loc+Sc⋅μ~1′=Loc+Sc(γ+ψ0(c))\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\left(\gamma+\psi_{0}\left(c\right)\right)

Varianza paramétrica

Variance(X)=Sc2⋅(μ~2′−μ~1′2)=Sc2(π26+ψ1(c))\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=ψ2(c)+2ζ(3)(π26+ψ1(c))3/2\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\frac{\psi_{2}\left(c\right)+2\zeta\left(3\right)}{\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)^{3/2}}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=(π415+ψ3(c))(π26+ψ1(c))2\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=\frac{\left(\frac{\pi^{4}}{15}+\psi_{3}\left(c\right)\right)}{\left(\frac{\pi^{2}}{6}+\psi_{1}\left(c\right)\right)^{2}}

Mediana paramétrica

Median(X)=Loc−Scln⁡(21/c−1)\mathrm{Median}(X)=\text{Loc}-\text{Sc}\ln\left(2^{1/c}-1\right)

Moda paramétrica

Mode(X)=Loc+Scln⁡(c)\mathrm{Mode}(X)=\text{Loc}+\text{Sc}\ln\left(c\right)

Información y definiciones adicionales

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
γ:Euler-Mascheroni constant=0.5772156649\gamma:\text{Euler-Mascheroni constant}=0.5772156649
ψ0(x):Digamma function\psi_{0}\left(x\right):\text{Digamma function}
ψn(x):Polygamma function of order n∈N\psi_{n}\left(x\right):\text{Polygamma function of order }n\in\mathbb{N}