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Kumaraswamy distribution

What it describes

The Kumaraswamy distribution is a bounded model for variables on an interval. Its practical appeal is an elementary CDF and quantile function, which makes simulation and calculation easy without inverting the incomplete Beta function.

In Phitter the support is [min,max]. alpha and beta control endpoint behaviour and allow J-shaped, reverse-J, uniform, or interior-peaked profiles.

History and names

P. Kumaraswamy introduced it in 1980 for doubly bounded processes, motivated by hydrological variables such as normalized levels and flows.

How it connects to other distributions

It shares Beta’s support and goals but is not the same family. It is used as a base for generated distributions, including exponentiated Kumaraswamy; alpha=beta=1 gives uniform.

Where it appears

  • normalized hydrological and environmental variables
  • bounded proportions and rates where an explicit CDF is valuable

Fitting cautions

Computational convenience is not statistical superiority; compare shape, quantiles, and predictive fit against Beta.

A hydrological alternative to Beta

Poondi Kumaraswamy introduced this distribution in 1980 for bounded hydrological processes. Like Beta, it can produce increasing, decreasing, unimodal, and U-shaped densities, but both its CDF and quantile use elementary operations.

That tractability helps simulation and some hierarchical models. It does not make Kumaraswamy and Beta identical: they differ especially in tail behaviour near zero and one. A simple power transformation reveals another connection, turning part of the construction into a Beta law with one shape equal to one. For data containing exact endpoint values, neither continuous density alone assigns probability mass to those points.

Decision guide

A good candidate when: a bounded continuous proportion is modelled and explicit CDF and quantiles are valuable for simulation or regression.

Compare it with: Beta. Inspect both endpoints: similar central shapes can assign different probabilities near zero and one; exact boundary values require another model.

References

Kumaraswamy Distribution: equations and calculator

Distribution defintion

X∼Kumaraswamy(α,β,min,max)X\sim\mathrm{Kumaraswamy}\left(\alpha,\beta,\text{min},\text{max}\right)

Distribution domain

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

Parameters domain and parameters constraints

α∈R+,β∈R+,min∈R,max∈R\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Cumulative distribution function

FX(x)=1−(1−z(x)α)βF_{X}\left(x\right)=1-(1-z(x)^\alpha)^\beta

Probability density function

fX(x)=αβz(x)α−1(1−z(x)α)β−1f_{X}\left(x\right)=\alpha \beta z(x)^{\alpha-1}(1-z(x)^\alpha)^{\beta-1}

Percent point function/Sample

FX−1(u)=min+(max−min)×(1−(1−u)1β)1αF^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\times (1-(1-u)^\frac{1}{\beta})^\frac{1}{\alpha}

Non-central parametric moments

μ~k′=E[X~k]=∫01xkfX~(x)dx=βBeta(1+kα,β)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\int_{0}^{1}x^{k}f_{\tilde{X}}\left(x\right)dx=\beta \text{Beta}(1+\frac{k}{\alpha},\beta)

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)×(1−2−1/b)1/a\mathrm{Median}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times\left(1-2^{-1/b}\right)^{1/a}

Parametric mode

Mode(X)=min+(max−min)×(a−1ab−1)1/a\mathrm{Mode}(X)=\text{min}+\left(\text{max}-\text{min}\right)\times\left(\frac{a-1}{ab-1}\right)^{1/a}

Additional information and definitions

X~∼Kumaraswamy(α,β,0,1)\tilde{X}\sim\mathrm{Kumaraswamy}\left(\alpha,\beta,0,1\right)
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}
Beta(x,y):Beta function\text{Beta}\left(x,y\right):\text{Beta function}