PLAYGROUND

ECUACIONES DISTRIBUCIÓN GAMMA GENERALIZADA 4P

Definición de distribución

X∼GeneralizedGamma4P(a,d,p,Loc)X\sim\mathrm{GeneralizedGamma_{4P}}\left(a,d,p,\text{Loc}\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

a∈R+,d∈R+,p∈R+,Loc∈Ra\in\mathbb{R}^{+},d\in\mathbb{R}^{+},p\in\mathbb{R}^{+},\text{Loc}\in\mathbb{R}

Función de distribución acumulada

FX(x)=P(d/p,((x−Loc)/a)p)=γ(d/p,((x−Loc)/a)p)Γ(d/p)F_{X}\left(x\right)=\text{P}(d/p,((x-\text{Loc})/a)^p)=\frac{\gamma(d/p,((x-\text{Loc})/a)^p)}{\Gamma(d/p)}

Función de densidad de probabilidad

fX(x)=p/adΓ(d/p)(x−Loc)d−1e−((x−Loc)/a)pf_{X}\left(x\right)=\frac{p/a^d}{\Gamma(d/p)} (x-\text{Loc})^{d-1}e^{-((x-\text{Loc})/a)^p}

Función de punto percentil

FX−1(u)=Loc+aP−1(dp,u)1pF^{-1}_{X}\left(u\right)=\text{Loc}+a\text{P}^{-1}\left(\frac{d}{p},u\right)^{\frac{1}{p}}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=akΓ(d+kp)Γ(dp)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=a^k\frac{\Gamma (\frac{d+k}{p})}{\Gamma(\frac{d}{p})}

Media paramétrica

Mean(X)=Loc+μ~1′\mathrm{Mean}(X)=\text{Loc}+\tilde{\mu}'_{1}

Varianza paramétrica

Variance(X)=μ~2′−μ~1′2\mathrm{Variance}(X)=\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1}

Coeficiente de asimetría paramétrico

Skewness(X)=μ~3′−3μ~2′μ~1′+2μ~1′3(μ~2′−μ~1′2)1.5\mathrm{Skewness}(X)=\frac{\tilde{\mu}'_{3}-3\tilde{\mu}'_{2}\tilde{\mu}'_{1}+2\tilde{\mu}'^{3}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{1.5}}

Curtosis paramétrica

Kurtosis(X)=μ~4′−4μ~1′μ~3′+6μ~1′2μ~2′−3μ~1′4(μ~2′−μ~1′2)2\mathrm{Kurtosis}(X)=\frac{\tilde{\mu}'_{4}-4\tilde{\mu}'_{1}\tilde{\mu}'_{3}+6\tilde{\mu}'^{2}_{1}\tilde{\mu}'_{2}-3\tilde{\mu}'^{4}_{1}}{(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})^{2}}

Mediana paramétrica

Median(X)=Loc+aP−1(dp,12)1p\mathrm{Median}(X)=\text{Loc}+a\text{P}^{-1}\left(\frac{d}{p},\frac{1}{2}\right)^{\frac{1}{p}}

Moda paramétrica

Mode(X)=Loc+a(d−1p)1pif d>1\mathrm{Mode}(X)=\text{Loc}+a\left(\frac{d-1}{p}\right)^{\frac{1}{p}} \quad \text{if } d>1

Información y definiciones adicionales

X~∼GeneralizedGamma(a,d,p)\tilde{X}\sim\mathrm{GeneralizedGamma}\left(a,d,p\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
a:Scale parametera:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
P(a,x)=γ(a,x)Γ(a):Regularized lower incomplete gamma function\text{P}\left(a,x\right)=\frac{\gamma(a,x)}{\Gamma(a)}:\text{Regularized lower incomplete gamma function}
P−1(a,u):Inverse of regularized lower incomplete gamma function\text{P}^{-1}\left(a,u\right):\text{Inverse of regularized lower incomplete gamma function}
γ(a,x):Lower incomplete gamma function\gamma\left(a,x\right):\text{Lower incomplete gamma function}
Γ(x):Gamma function\Gamma\left(x\right):\text{Gamma function}