PLAYGROUND

ECUACIONES DISTRIBUCIÓN DAGUM 4P

Definición de distribución

X∼Dagum4P(a,b,p,Loc)X\sim\mathrm{Dagum_{4P}}\left(a,b,p,\text{Loc}\right)

Dominio de distribución

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

Dominio y restricciones de parámetros

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

Función de distribución acumulada

FX(x)=(1+(x−Locb)−a)−pF_{X}\left(x\right)={\left(1+{\left(\frac{x-\text{Loc}}{b}\right)}^{-a}\right)}^{-p}

Función de densidad de probabilidad

fX(x)=apx−Loc((x−Locb)ap((x−Locb)a+1)p+1)f_{X}\left(x\right)=\frac{a p}{x-\text{Loc}}\left(\frac{(\tfrac{x-\text{Loc}}{b})^{a p}}{\left((\tfrac{x-\text{Loc}}{b})^a+1\right)^{p+1}}\right)

Función de punto percentil

FX−1(u)=Loc+b(u−1/p−1)−1/aF^{-1}_{X}\left(u\right)=\text{Loc}+b(u^{-1/p}-1)^{-1/a}

Momentos paramétricos no centrados

μ~k′=E[X~k]=∫0∞xkfX~(x)dx=pbk⋅Beta(ap+ka,a−ka)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{\infty}x^{k}f_{\tilde{X}}\left(x\right)dx=pb^{k}\cdot \text{Beta}\left(\frac{ap+k}{a},\frac{a-k}{a}\right)

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+b(−1+21p)−1a\mathrm{Median}(X)=\text{Loc}+b{\left(-1+2^{\tfrac{1}{p}}\right)}^{-\tfrac{1}{a}}

Moda paramétrica

Mode(X)=Loc+b(ap−1a+1)1a\mathrm{Mode}(X)=\text{Loc}+b{\left(\frac{ap-1}{a+1}\right)}^{\tfrac{1}{a}}

Información y definiciones adicionales

Xˉ∼Dagum(a,b,p)\bar{X}\sim\mathrm{Dagum}\left(a,b,p\right)
Loc:Location parameter\text{Loc}:\text{Location parameter}
b:Scale parameterb:\text{Scale parameter}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}