PLAYGROUND

ECUACIONES DISTRIBUCIÓN JOHNSON SU

Definición de distribución

X∼JohnsonSU(ξ,λ,γ,δ)X\sim\mathrm{JohnsonSU}\left(\xi,\lambda,\gamma,\delta\right)

Dominio de distribución

x∈(−∞,∞)x\in\left(-\infty,\infty\right)

Dominio y restricciones de parámetros

ξ∈R,λ∈R+,γ∈R,δ∈R+\xi\in\mathbb{R},\lambda\in\mathbb{R}^{+},\gamma\in\mathbb{R},\delta\in\mathbb{R}^{+}

Función de distribución acumulada

FX(x)=Φ(γ+δsinh⁡−1(z(x)))F_{X}\left(x\right)=\Phi\left(\gamma+\delta\sinh^{-1}(z(x))\right)

Función de densidad de probabilidad

fX(x)=δλ2πz(x)2+1exp⁡[−12(γ+δsinh⁡−1(z(x)))2]f_{X}\left(x\right)=\frac{\delta}{\lambda\sqrt{2\pi}\sqrt{z(x)^2+1}}\exp\left[-\frac{1}{2}\left(\gamma+\delta\sinh^{-1}(z(x))\right)^2\right]

Función de punto percentil

FX−1(u)=λsinh⁡(Φ−1(u)−γδ)+ξF^{-1}_{X}\left(u\right)=\lambda\sinh\left(\frac{\Phi^{-1}(u)-\gamma}{\delta}\right)+\xi

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)=μ1′=ξ−λexp⁡δ−22sinh⁡(γδ)\mathrm{Mean}(X)=\mu'_{1}=\xi-\lambda \exp\frac{\delta^{-2}}{2} \sinh\left(\frac{\gamma}{\delta}\right)

Varianza paramétrica

Variance(X)=μ2′−μ1′2=λ22(exp⁡(δ−2)−1)(exp⁡(δ−2)cosh⁡(2γδ)+1)\mathrm{Variance}(X)=\mu'_{2}-\mu'^{2}_{1}=\frac{\lambda^2}{2} (\exp(\delta^{-2})-1)\left(\exp(\delta^{-2}) \cosh\left(\frac{2\gamma}{\delta}\right) +1\right)

Coeficiente de asimetría paramétrico

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=−λ3eδ−2(eδ−2−1)2(eδ−2)(eδ−2+2)sinh⁡(3γδ)+3sinh⁡(2γδ))4Variance(X)1.5\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=-\frac{\lambda^{3}\sqrt{e^{\delta^{-2}}}(e^{\delta^{-2}}-1)^{2}(e^{\delta^{-2}})(e^{\delta^{-2}}+2)\sinh(\frac{3\gamma}{\delta})+3\sinh(\frac{2\gamma}{\delta}))}{4\mathrm{Variance}(X)^{1.5}}

Curtosis paramétrica

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=λ4(eδ−2−1)2(K1+K2+K3)8Variance(X)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{\lambda^{4}(e^{\delta^{-2}}-1)^{2}(K_{1}+K_{2}+K_{3})}{8\mathrm{Variance}(X)^{2}}

Mediana paramétrica

Median(X)=ξ+λsinh⁡(−γδ)\mathrm{Median}(X)=\xi+\lambda \sinh\left(-\frac{\gamma}{\delta}\right)

Moda paramétrica

Mode(X)=arg⁡max⁡xfX(x)\mathrm{Mode}(X)=\arg\max_{x}f_{X}\left(x\right)

Información y definiciones adicionales

ξ:Location parameter\xi:\text{Location parameter}
λ:Scale parameter\lambda:\text{Scale parameter}
z(x)=(x−ξ)/λz\left(x\right)=\left(x-\xi\right)/\lambda
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
K1=(eδ−2)2((eδ−2)4+2(eδ−2)3+3(eδ−2)2−3)cosh⁡(4γδ)K_{1}=\left(e^{\delta^{-2}}\right)^{2}\left(\left(e^{\delta^{-2}}\right)^{4}+2\left(e^{\delta^{-2}}\right)^{3}+3\left(e^{\delta^{-2}}\right)^{2}-3\right)\cosh\left(\frac{4\gamma}{\delta}\right)
K2=4(eδ−2)2((eδ−2)+2)cosh⁡(3γδ)K_{2}=4\left(e^{\delta^{-2}}\right)^{2}\left(\left(e^{\delta^{-2}}\right)+2\right)\cosh\left(\frac{3\gamma}{\delta}\right)
K3=3(2(eδ−2)+1)K_{3}=3\left(2\left(e^{\delta^{-2}}\right)+1\right)