PLAYGROUND

ECUACIONES DISTRIBUCIÓN MCDONALD

Definición de distribución

X∼McDonald(a,b,c,min,max)X\sim\mathrm{McDonald}\left(a,b,c,\text{min},\text{max}\right)

Dominio de distribución

x∈(min,max)x\in\left(\text{min},\text{max}\right)

Dominio y restricciones de parámetros

a∈R+,b∈R+,c∈R+,min∈R,max∈Ra\in\mathbb{R}^{+},b\in\mathbb{R}^{+},c\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Función de distribución acumulada

FX(x)=Iz(x)c(a,b)F_{X}\left(x\right)=I_{z(x)^{c}}\left(a,b\right)

Función de densidad de probabilidad

fX(x)=c(max−min)Beta(a,b) z(x)ac−1(1−z(x)c)b−1f_{X}\left(x\right)=\frac{c}{\left(\text{max}-\text{min}\right)\text{Beta}(a,b)}\,z(x)^{ac-1}\left(1-z(x)^{c}\right)^{b-1}

Función de punto percentil

FX−1(u)=min+(max−min)[Iu−1(a,b)]1/cF^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\left[I^{-1}_{u}\left(a,b\right)\right]^{1/c}

Momentos paramétricos no centrados

μ~k′=E[X~k]=Beta ⁣(a+kc,b)Beta(a,b)\tilde{\mu}'_{k}=E[\tilde{X}^{k}]=\frac{\text{Beta}\!\left(a+\tfrac{k}{c},b\right)}{\text{Beta}(a,b)}

Media paramétrica

Mean(X)=min+(max−min)×μ~1′\mathrm{Mean}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times \tilde{\mu}'_{1}

Varianza paramétrica

Variance(X)=(max−min)2(μ~2′−μ~1′2)\mathrm{Variance}(X)=\left(\text{max}-\text{min}\right)^{2}(\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)=min+(max−min)[I0.5−1(a,b)]1/c\mathrm{Median}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left[I^{-1}_{0.5}\left(a,b\right)\right]^{1/c}

Moda paramétrica

Mode(X)=min+(max−min)(ac−1ac+bc−c−1)1/c\mathrm{Mode}(X)=\text{min}+\left(\text{max}-\text{min}\right)\left(\frac{ac-1}{ac+bc-c-1}\right)^{1/c}

Información y definiciones adicionales

X~∼McDonald(a,b,c,0,1)\tilde{X}\sim\mathrm{McDonald}\left(a,b,c,0,1\right)
If Xc∼Beta(a,b) then X∼McDonald(a,b,c)\text{If }X^{c}\sim\mathrm{Beta}(a,b)\text{ then }X\sim\mathrm{McDonald}(a,b,c)
z(x)=(x−min)/(max−min)z\left(x\right)=\left(x-\text{min}\right)/\left(\text{max}-\text{min}\right)
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Ix(a,b):Regularized incomplete beta functionI_{x}(a,b):\text{Regularized incomplete beta function}
Iu−1(a,b):Inverse of regularized incomplete beta functionI^{-1}_{u}(a,b):\text{Inverse of regularized incomplete beta function}
Beta(a,b):Beta function\text{Beta}(a,b):\text{Beta function}
Mode formula valid when ac>1 and b>1\text{Mode formula valid when } ac>1 \text{ and } b>1