PLAYGROUND

McDonald distribution

Why it exists

The McDonald distribution is a bounded family generated by applying a power to a Beta variable. It uses five parameters to control endpoints, shape, and concentration within a finite interval.

Alternative names

It is also called the McDonald distribution or generalized Beta type I. Do not confuse GB1, bounded, with GB2, which has positive unbounded support.

Constructions and limits

At c=1 it becomes a rescaled Beta. GB1 belongs to the generalized-Beta tree, which also contains models related to Dagum, Burr, and loglogistic.

Applications

It can help when measurement fixes an interval but shape needs more degrees of freedom than Beta. In economic data, bounds and cross-population comparability matter as much as the curve.

Five parameters can overfit bounded samples; compare Beta and report endpoint uncertainty.

A Beta distribution with an extra lever

McDonald developed generalized Beta families for economic size data. In the bounded form used by Phitter, a power of relative position within [min,max] is fed into a Beta law. The extra parameter changes behaviour near the lower endpoint without forcing the upper one to move.

This distribution is also described as generalized Beta of the first kind. It should not be confused with GB2, which has unbounded positive support and a central role in income modelling. Five parameters can reproduce many silhouettes, but they also create practical equivalences. If Beta or Kumaraswamy explains the relevant quantiles, choosing the simpler curve improves both stability and communication.

Decision guide

A good candidate when: data are bounded and require more shape control than Beta or Kumaraswamy provide.

Compare it with: those simpler families using out-of-sample validation. Five parameters can mimic many curves and produce substantial endpoint uncertainty.

References

McDonald Distribution: equations and calculator

Distribution defintion

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

Distribution domain

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

Parameters domain and parameters constraints

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}

Cumulative distribution function

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

Probability density function

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}

Percent point function/Sample

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}

Non-central parametric moments

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)}

Parametric mean

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

Parametric variance

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})

Parametric skewness

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}}

Parametric kurtosis

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}}

Parametric median

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}

Parametric mode

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}

Additional information and definitions

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