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Exponentiated Kumaraswamy distribution

What it represents

The exponentiated Kumaraswamy distribution combines a bounded Kumaraswamy law with an additional power transformation. The extra layer can create decreasing, U-shaped, or peaked profiles while staying on [min,max].

Historical clue

The construction belongs to CDF-generated distribution families. It is more recent than the classical families and is studied as a flexible model for unit-interval data.

Relationships that clarify its use

At lambda=1 it reduces to Kumaraswamy. It shares Beta’s bounded-data motivation, but its standard CDF and quantiles can be expressed with elementary operations rather than the incomplete Beta function.

Data examples

  • normalized proportions and rates on (min,max)
  • bounded data whose skewness or endpoint concentration a simple Beta cannot capture

Modelling warning

Many parameters on a short interval can be weakly identified; compare Beta and Kumaraswamy on the same support as a minimum check.

Adding a power to an already explicit CDF

The exponentiated construction raises the Kumaraswamy CDF to a positive parameter. If that parameter is an integer, it can be interpreted as the maximum of several independent Kumaraswamy variables. At one, the original family is recovered.

The extra layer moves probability and permits bounded curves unavailable as easily from the two-shape version. It retains an explicit quantile function, which helps simulation. That convenience does not remove competition among parameters: several combinations may yield nearly identical central quantiles and differ only near the endpoints. For proportions, inspect both ends and compare Beta, Kumaraswamy, and McDonald before paying for three shapes that the data may not identify.

Decision guide

A good candidate when: bounded shapes unavailable to Beta or Kumaraswamy are required and the sample is large enough to identify the additional power parameter.

Compare it with: Beta and Kumaraswamy using out-of-sample validation. Extra flexibility is justified only when it improves relevant quantiles or decisions, not merely in-sample fit.

References

Exponentiated Kumaraswamy Distribution: equations and calculator

Distribution defintion

X∼ExponentiatedKumaraswamy(α,β,λ,min,max)X\sim\mathrm{ExponentiatedKumaraswamy}\left(\alpha,\beta,\lambda,\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+,λ∈R+,min∈R,max∈R\alpha\in\mathbb{R}^{+},\beta\in\mathbb{R}^{+},\lambda\in\mathbb{R}^{+},\text{min}\in\mathbb{R},\text{max}\in\mathbb{R}

Cumulative distribution function

FX(x)=[1−(1−z(x)α)β]λF_{X}\left(x\right)=\left[1-\left(1-z(x)^{\alpha}\right)^{\beta}\right]^{\lambda}

Probability density function

fX(x)=αβλmax−min z(x)α−1(1−z(x)α)β−1[1−(1−z(x)α)β]λ−1f_{X}\left(x\right)=\frac{\alpha\beta\lambda}{\text{max}-\text{min}}\,z(x)^{\alpha-1}\left(1-z(x)^{\alpha}\right)^{\beta-1}\left[1-\left(1-z(x)^{\alpha}\right)^{\beta}\right]^{\lambda-1}

Percent point function/Sample

FX−1(u)=min+(max−min)[1−(1−u1/λ)1/β]1/αF^{-1}_{X}\left(u\right)=\text{min}+\left(\text{max}-\text{min}\right)\left[1-\left(1-u^{1/\lambda}\right)^{1/\beta}\right]^{1/\alpha}

Non-central parametric moments

μ~k′=E[X~k]=λ∑j=0∞(−1)j(k/αj)Beta ⁣(jβ+1,λ)\tilde{\mu}'_{k}=E[\tilde{X}^k]=\lambda\sum_{j=0}^{\infty}(-1)^{j}\binom{k/\alpha}{j}\text{Beta}\!\left(\tfrac{j}{\beta}+1,\lambda\right)

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)=FX−1(0.5)\mathrm{Median}(X)=F^{-1}_{X}\left(0.5\right)

Parametric mode

Mode(X)=arg⁡max⁡x∈(min,max)fX(x)\mathrm{Mode}(X)=\arg\max_{x\in(\text{min},\text{max})} f_{X}(x)

Additional information and definitions

X~∼ExponentiatedKumaraswamy(α,β,λ,0,1)\tilde{X}\sim\mathrm{ExponentiatedKumaraswamy}\left(\alpha,\beta,\lambda,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}
(νj):Generalized binomial coefficient\binom{\nu}{j}:\text{Generalized binomial coefficient}